hart interactive –algebra 2/trigonometry mathematics … · · 2017-01-03algebra 2/trig •...

ALGEBRA 2/TRIG • MODULE 3

Hart Interactive –Algebra 2/Trigonometry

Mathematics Curriculum Table of Contents1

Trigonometric Functions Module Overview ................................................................................................................................................. 2

Unit 6: Trigonometry – Focus on Sine, Cosine and Tangent (F-IF.C.7e, F-TF.A.1, F-TF.A.2)

Unit 6 Vocabulary .................................................................................................................................. S.1

Lesson 1: Ferris Wheels—Tracking the Height of a Passenger Car ...................................................... S.7

Lesson 2: A Model Ferris Wheel ........................................................................................................ S.13

Lesson 3: The Height and Co-Height Functions of a Ferris Wheel ...................................................... S.19

Lesson 4: The Unit Circle..................................................................................................................... S.29

Lesson 5: From Circle-ometry to Trigonometry ................................................................................. S.35

Lesson 6: Extending the Domain of Sine and Cosine to All Real Numbers ......................................... S.47

Lesson 7: Why Call It Tangent? ........................................................................................................... S.55

Lesson 8: Secant and the Co-Functions .............................................................................................. S.67

Lesson 9: Graphing the Sine and Cosine Functions – Spaghetti Graphs ............................................ S.79

Lesson 10: Graphing the Sine and Cosine Functions – Looking at Patterns ...................................... S.83

Unit 7: Trig Identities and Transformations (F-IF.C.7e, F-TF.B.5, F-TF.C.8, S.ID.B.6a)

Unit 7 Vocabulary ................................................................................................................................ S.91

Lesson 11: Awkward! Who Chose the Number 360, Anyway? ......................................................... S.97

Lesson 12: Basic Trigonometric Identities from Graphs ................................................................... S.107

Lesson 13: Transforming the Graph of the Sine Function – A and ω .............................................. S.121

Lesson 14: Transforming the Graph of the Sine Function – h and k ................................................ S.127

Lesson 15: Transforming the Graph of the Sine Function – Putting It All Together ........................ S.133

Lesson 16: Tides, Sound Waves, and Stock Markets – Modeling with Sinusoidal Graphs ............... S.145

Lesson 17: Graphing the Tangent Function ...................................................................................... S.159

Lesson 18: What Is a Trigonometric Identity? .................................................................................. S.177

Lesson 19: Proving Trigonometric Identities .................................................................................... S.185

Lesson 20: Trigonometric Identity Proofs (Honors) ......................................................................... S.199

Lesson 21: Ferris Wheels—Using Parametric Functions to Model Cyclical Behavior (Honors) ....... S.209

1Each lesson is ONE day, and ONE day is considered a 45-minute period.

Module 3: Trigonometric Functions

1

This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG II-M2-TE-1.3.0-08.2015

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

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ALGEBRA II/TRIGONOMETRY

Hart Interactive – Algebra 2/Trig M3 Module Overview

Algebra II/Trigonometry • Module 3

Trigonometric Functions OVERVIEW Module 3 builds on your previous work with units (N-Q.A.1) and with functions (F-IF.A.1, F-IF.A.2, F-IF.B.4, F-IF.C.7e, F-BF.A.1, F-BF.B.3) from Algebra I and with trigonometric ratios and circles (G-SRT.C.6, G-SRT.C.7, G-SRT.C.8) from high school Geometry. Included in Unit 6 is preparation for extension standard F-TF.A.3. Extension standard F-TF.C.9 is also discussed in Unit 7 as preparation for the Precalculus and Advanced Topics course.

Unit 6 starts by asking you to graph the height of a passenger car on a Ferris wheel as a function of how much rotation it has undergone and uses that study to help you define the sine, cosine, and tangent functions as functions from all (or most) real numbers to the real numbers. A precise definition of sine and cosine (as well as tangent and the co-functions) is developed using transformational geometry. This precision leads to a discussion of a mathematically natural unit of measurement for angle measures, a radian, and you’ll begin to build fluency with values of sine, cosine, and tangent at

𝜋𝜋6

, 𝜋𝜋4

, 𝜋𝜋3

, 𝜋𝜋2

, 𝜋𝜋, etc. The topic concludes with you graphing the sine and cosine functions and noticing various aspects of the graph, which you’ll write down as simple trigonometric identities.

In Unit 7, you will make sense of periodic phenomena as you model them with trigonometric functions. You’ll identify the periodicity, midline, and amplitude from graphs of data and use them to construct sinusoidal functions that model situations from both the biological and physical sciences. You’ll extend the concept of polynomial identities to trigonometric identities and prove simple trigonometric identities such as the Pythagorean identity; these identities are then used to solve problems.

Module 3: Trigonometric Functions

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Hart Interactive – Algebra 2/Trigonometry

Unit 6 Vocabulary

Knowledge Rating*

Lesson Vocabulary

Terms Definition Picture/Example/Notation

1 Rotation A rotation has a central point that stays fixed and everything else moves ____________ that point in a circle.

1 Counter-clockwise

Counter-clockwise moves in a direction opposite to that in which the ________ of a clock move.

1 Clockwise Clockwise moves in the direction of the ________ of a clock.

1 Circle A 2-dimensional shape made by drawing a curve that is always the same ____________ from a center.

1 Radius The ____________ from the center of a circle to the outside edge.

1 Diameter A straight line going through the ________ of a circle connecting two points on the circumference.

1 360°

Rotation (Turn)

360° rotation is turning around until you point in the ________ direction again.

Knowledge Rating: N = I have no knowledge of the word. S = I’ve seen the word, but I’m not sure what it means. U = I understand this word and can use it correctly.

Module 3: Trigonometric Functions Unit 6: Trigonometry – Focus on Sine, Cosine and Tangent

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This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015






ALGEBRA 2/TRIG

M3 Unit 6 Vocabulary Hart Interactive – Algebra 2/Trigonometry

Knowledge Rating*

Lesson Vocabulary


2 Degree A unit of ________________ of a plane ________ , so that a full rotation is 360 degrees. The symbol for degrees is °.

2 Degrees of Rotation

The ________ of rotation in terms of degrees. If the degrees are positive, the rotation is counterclockwise; if they are negative, the rotation is clockwise.

2 Domain The set of all possible ________ values (commonly the "x" variable), which produce a valid output from a particular function.

2 Range

The range is the set of all possible ________ values (commonly the “y” variable), which result from using a particular function.

2 Periodic Function

Periodic function is a function that ____________ its values in regular intervals or periods.

3 Co-Height The ________________ distance with respect to the vertical line through the center (y-axis).

3 Maximum The height of the function that is ____________ than (or equal to) the height anywhere else in that interval.

3 Minimum The ____________ value of a function in that interval.



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ALGEBRA 2/TRIG


Knowledge Rating*

Lesson Vocabulary


4 Unit Circle The unit circle has a center at the origin (0,0) and a ________ of 1.

4 Quadrantal Angles

Angles in the standard position, that are a multiple of ____°, where the terminal side lies on the x- or y-axis.

4 Special Right Triangles

A special right triangle is a right triangle whose sides are in a particular _______. The most common are the 30°-60°-90° and the 45°-45°-90° triangles.

5 Sine The length of the ____________ side divided by the length of the ____________ in a right triangle.

5 Cosine The length of the ____________ side divided by the length of the ____________ in a right triangle.

5 Initial Ray The ray where measurement of an angle ________.

5 Terminal Ray The ray where measurement of an angle ________.

5 Reference Angle

The smallest angle between the terminal side and the ____-axis.



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http://www.mathopenref.com/trigstandardposition.html

http://www.mathopenref.com/trigterminalside.html

ALGEBRA 2/TRIG


Knowledge Rating*

Lesson Vocabulary


7 Tangent

The length of the ____________ side divided by the length of the ____________ side in a right triangle.

7 Secant

The length of the ____________ divided by the length of the ____________ side in a right triangle. This is the reciprocal of a cosine.

8 Co-Functions The function of the ____________ of a given angle.

8 Cosecant

The length of the ____________ divided by the length of the ____________ side in a right triangle. This is the reciprocal of sine.

8 Cotangent

The length of the ____________ side divided by the length of the ____________ side in s right triangle. This is the reciprocal of tangent.

9 Period The ____________ from one peak to the next of a periodic function to complete one full cycle.

10 Sinusoidal Wave

Mathematical curve that describes a smooth ____________ oscillation.



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ALGEBRA 2/TRIG


Name: ____________________________________________________________________ Period: ______

Use one of the words in the Unit 6 set of vocabulary to complete this Frayer diagram. Your teacher may assign you a word so ask before you start. Your word should be written in the middle circle. Be prepared to share out with the class.


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ALGEBRA 2/TRIG



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Hart Interactive – Algebra 2/Trig M3 Lesson 1 ALGEBRA II/TRIGONOMETRY

Lesson 1: Ferris Wheels—Tracking the Height of a Passenger Car

Exploratory Challenge 1: The Height of a Ferris Wheel Car

George Ferris built the first Ferris wheel in 1893 for the World’s Columbian Exhibition in Chicago. It had 30 passenger cars, was 264 feet tall and rotated once every 9 minutes when all the cars were loaded. The ride cost $0.50.

Source: The New York Times/Redux

Discussion 1. Consider how the height of a single passenger car is changing as the car rotates around the wheel. A. What quantities are changing as the Ferris wheel rotates? B. How does the height of a single car change as the car rotates around the wheel?

Lesson 1: Ferris Wheels—Tracking the Height of a Passenger Car Unit 6: Trigonometry – Focus on Sine, Cosine and Tangent

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2. A. Create a sketch of the height of a passenger car on the original Ferris wheel as that car rotates around the wheel 4 times. List any assumptions that you are making as you create your model.

B. What type of function would best model this situation?

Discussion

3. How did you decide where to locate your points on this graph? 4. How could you make your model more precise? 5. Is a passenger car ever at the same height during its rotation? How do you know?


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6. Where did you choose to start your car and why? 7. Do you think the height is changing in a linear fashion? Why or why not? 8. What patterns did you notice in the way the height of the passenger car was changing? 9. How does the range of the graph of the Ferris wheel height function relate to the physical features of the

wheel?

Extending Your Thinking – Exercises 10 – 14

10. Suppose a Ferris wheel has a diameter of 150 feet. From your viewpoint, the Ferris wheel is rotating counterclockwise. We will refer to a rotation through a full 360° as a turn. A. Create a sketch of the height of a car that starts at the bottom of the wheel and continues for two

turns.

B. Explain how the features of your graph relate to this situation.


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11. Suppose a Ferris wheel has a diameter of 150 feet. From your viewpoint, the Ferris wheel is rotating counterclockwise. A. Your friends board the Ferris wheel, and the ride continues boarding passengers. Their car is in the

three o’clock position when the ride begins. Create a sketch of the height of your friends’ car for two turns.

B. Explain how the features of your graph relate to this situation.

12. How would your sketch change if the diameter of the wheel changed?

13. If you translated the sketch of your graph down by the radius of the wheel, what would the 𝑥𝑥-axis represent in this situation?

14. How could we create a more precise sketch?


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Homework Problem Set 1. A small pebble is lodged in the tread of a tire with radius

25 cm. Sketch the height of the pebble above the ground as the tire rotates counterclockwise through 5 turns. Start your graph when the pebble is at the 9 o’clock position.

2. The graph you created in Problem 1 represents a function. a. Describe how the function and its graph would change if the tire’s radius was 24 inches instead of

25 cm.

b. Describe how the function and its graph would change if the wheel was turning in the opposite direction.

c. Describe how the function and its graph would change if we started the graph when the pebble was at ground level.


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3. Justice believes that the height of a Ferris wheel passenger car is best modeled with a piecewise linear function. Make a convincing argument why a piecewise linear function is NOT a good model for the height of a car on a rotating Ferris wheel.

SPIRAL REVIEW – Proportions Write and solve a proportion for each problem below. 4. Diana spends 17 hours in a 2-week period practicing the piano. How many hours does she practice in 7

weeks? 5. A car is traveling 924 feet in 1 minute. How fast is this in feet per hour? 6. Allen makes $15.00 per hour. Last week he made $142.50. How many hours did Allen work last week? 7. A 5-pound bag of oranges cost $7.25. How much will a 3-pound bag cost? 8. A recipe calls for 1 ¾ cups of flour. Mimsy wants to triple the batch. How much flour will she need?


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Lesson 2: A Model Ferris Wheel

In Lesson 1 you created a graph of the Ferris wheel’s height over time and considered changes to the diameter. In this lesson, you and your partner will create a paper model of a Ferris wheel and continue your investigation into the graphs of the height.

Exploratory Challenge: The Paper Plate Model

1. Suppose that your friends board the Ferris wheel near the end of the boarding period and the ride begins when their car is in the three o’clock position as shown. A. The point on the circle below represents the passenger car in the 3 o’clock position. Since this is the

beginning of the ride, consider this position to be the result of rotating by 0°. Mark the diagram below to estimate the location of the Ferris wheel passenger car every 15 degrees.

B. How many degrees will your friends have passed through once they have completed one complete rotation?

Paper plate

What you’ll measure at

0°.

Lesson 2: A Model Ferris Wheel Unit 6: Trigonometry – Focus on Sine, Cosine and Tangent

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You will need: a ruler, protractor, colored pencils or pens, small paper plate, cardstock and metal brad fastener

2. A. Your model will be a paper plate mounted on a sheet of cardstock, where the lower edge of the paper represents the ground. You’ll use a ruler and protractor to measure the height of a Ferris wheel car above the ground for various amounts of rotation.

B. Using the physical model you created with your group, record your measurements in the table, and then graph the ordered pairs (rotation, height) on the coordinate grid shown below. Provide appropriate labels on the axes.

Rotation (degrees)

Height (cm)

Rotation (degrees)

Height (cm)

Rotation (degrees)

Height (cm)

Rotation (degrees)

Height (cm)

0 105 210 315

15 120 225 330

30 135 240 345

45 150 255 360

60 165 270

75 180 285

90 195 300


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3. Explain how the features of your graph relate to the paper plate model you created.

4. If the paper plate model was scaled so that 1 cm on the plate represented 5 ft. on a real Ferris wheel, what is the diameter of the wheel?

5. How high above the ground is the lowest point on the Ferris wheel? 6. Why isn’t the diameter of the Ferris wheel the same as the maximum value on your graph?

7. How does a function like the one that represents the height of a passenger car on a Ferris wheel differ from other types of functions you have studied such as linear, polynomial, and exponential functions?


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8. What is the domain of your Ferris wheel height function? What is the range?

9. The Ferris wheel height function an example of a periodic function. A periodic function is defined as a

function that repeats its values in regular intervals or periods.

A. Write a definition of periodic function in your own words.

B. What other situations might be modeled by a periodic function?


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Homework Problem Set For Problems 1 – 4, suppose that a Ferris wheel is 40 feet in diameter and rotates counterclockwise. When a passenger car is at the bottom of the wheel, it is located 2 feet above the ground. 1. A. Sketch a graph of a function that represents the height of a passenger car that starts at the 3 o’clock

position on the wheel for one turn.

B. The sketch you created in Part A represents a graph of a function. What is the domain of the function? What is the range?

2. A. Sketch a graph of a function that represents the height of a passenger car that starts at the top of the

wheel for one turn.

B. The sketch you created in Part A represents a graph of a function. What is the domain of the function? What is the range?


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3. Describe how the graph of the function in Problem 1 would change if you sketched the graph for two turns.

4. Describe how the function in Problem 1 and its graph would change if the Ferris wheel had a diameter of

60 feet.

SPIRAL REVIEW – Similar Triangles For each set of similar triangles below, determine the missing side lengths. Drawings are not to scale. Use the similarity statements.

5. ∆ABE ∼ ∆DBC 6. ∆UTV ∼ ∆WVS 7. ∆IGH ∼ ∆LJK

SPIRAL REVIEW – SohCahToa 8. Given the following right triangle △ 𝐴𝐴𝐴𝐴𝐴𝐴 with 𝑚𝑚∠𝐴𝐴 = 𝜃𝜃°, find sin(𝜃𝜃°) and cos(𝜃𝜃°).

x

w

x

w 66

x

w 25


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Lesson 3: The Height and Co-Height Functions of a Ferris Wheel

Opening Exercise

1. Suppose a Ferris wheel has a radius of 50 feet. We will measure the height of a passenger car that starts in the 3 o’clock position with respect to the horizontal line through the center of the wheel. That is, we consider the height of the passenger car at the outset of the problem (that is, after a 0° rotation) to be 0 feet.

A. Mark the diagram to show the position of a passenger car at 30-degree intervals as it rotates counterclockwise around the Ferris wheel.

B. Sketch the graph of the height function of the passenger car for one turn of the wheel. Provide appropriate labels on the axes.

C. Explain how you can identify the radius of the wheel from the graph in Part B.

D. If the center of the wheel is 55 feet above the ground, how high is the passenger car above the ground when it is at the top of the wheel?

Lesson 3: The Height and Co-Height Functions of a Ferris Wheel Unit 6: Trigonometry – Focus on Sine, Cosine and Tangent

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Recall that we modeled the height of a passenger car as a function of degrees as the car rotated counterclockwise from the car’s starting point at a certain point on the wheel—either the bottom of the wheel or at the 3 o’clock position. 2. Is there another measurement that we can model as a function of degrees rotated counterclockwise from

the car’s starting position? In the Opening Exercise, we changed how we measure the height of a passenger car on the Ferris wheel, and we now consider the height to be the vertical displacement from the center of the wheel. Points near the top of the wheel have a positive height, and points near the bottom have a negative height. That is, we measure the height as the vertical distance from a horizontal line through the center of the wheel. 3. With this in mind, how should we measure the horizontal distance? We will refer to the horizontal displacement of a passenger car from the vertical line through the center of the wheel as the co-height of the car. 4. A. Where is the car when the co-height is zero?

B. How can we assign positive and negative values to the co-height?

5. Using our Opening Exercise, what is the starting value of the co-height?


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6. Now suppose, that the passenger car has rotated 90 degrees counterclockwise from its initial position of 3 o’clock on the wheel. What is the co-height of the car in this position?

7. Is there a maximum value of the co-height of a passenger car? Is there a

minimum value of the co-height?

8. Each point 𝑃𝑃1, 𝑃𝑃2, … 𝑃𝑃8 on the circle in the diagram to the right represents a passenger car on a Ferris wheel. A. Draw segments that represent the co-height of each car. B. Which cars have a positive co-height? Which cars have a negative

co-height?

C. List the points in order of increasing co-height; that is, list the point with the smallest co-height first and the point with the largest co-height last.

9. Suppose that the radius of a Ferris wheel is 100 feet and the wheel rotates counterclockwise through one turn. Define a function that measures the co-height of a passenger car as a function of the degrees of rotation from the initial 3 o’clock position. A. What is the domain of the co-height function?

B. What is the range of the co-height function?

C. How does changing the wheel’s radius affect the domain and range of the co-height function?

10. For a Ferris wheel of radius 100 feet going through one turn, how do the domain and range of the height function compare to the domain and range of the co-height function? Is this true for any Ferris wheel?


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Exploratory Challenge: The Paper Plate Model, Revisited

Use your Ferris wheel model from Lesson 2 to measure the height and co-height of a Ferris wheel car at various amounts of rotation, measured with respect to the horizontal and vertical lines through the center of the wheel. Suppose that your friends board the Ferris wheel near the end of the boarding period, and the ride begins when their car is in the three o’clock position.

11. Using the physical model you created with your group, record your measurements in the table, and then

graph each of the two sets of ordered pairs (rotation angle, height) and (rotation angle, co-height) on separate coordinate grids. Provide appropriate labels on the axes.

Rotation (degrees)

Height (cm)

Co-Height (cm)

Rotation (degrees)

Height (cm)

Co-Height (cm)

Rotation (degrees)

Height (cm)

Co-Height (cm)

0 135 255

15 150 270

30 165 285

45 180 300

60 195 315

75 210 330

90 225 345

105 240 360

120


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Height as a Function of Degrees of Rotation

Co-Height as a Function of Degrees of Rotation


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Closing

12. Why do you think we named the new function the co-height?

13. How are the graphs of these two functions alike? How are they different?

14. What does a negative value of the height function tell us about the location of the passenger car at various positions around a Ferris wheel? What about a negative value of the co-height function?

15. What do the zeros of the graph of the co-height function represent in this situation? 16. What does the vertical intercept of the graph of the co-height function represent in this situation? 17. How are the graphs of the height and co-height functions related to each other?

Lesson Summary In this lesson, we defined a new measurement on the Ferris wheel, the co-height.

Periodic graphs, like the ones for the Ferris wheel, repeat their pattern. Another real-life example of a periodic function is the height of the seat of a swing as a child swings in it.


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Homework Problem Set 1. The Seattle Great Wheel, with an overall height of

175 feet, was the tallest Ferris wheel on the West Coast at the time of its construction in 2012. For this exercise, assume that the diameter of the wheel is 175 feet. a. Create a diagram that shows the position of a

passenger car on the Great Wheel as it rotates counterclockwise at 45-degree intervals.

b. On the same set of axes, sketch graphs of the height and co-height functions for a passenger car starting at the 3 o’clock position on the Great Wheel and completing one turn.


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c. Discuss the similarities and differences between the graph of the height function and the graph of the co-height function.

d. Explain how you can identify the radius of the wheel from either graph.

2. In 2014, the High Roller Ferris wheel opened in Las Vegas, dwarfing the Seattle Great Wheel with a diameter of 520 feet. Sketch graphs of the height and co-height functions for one complete turn of the High Roller.

3. Consider a Ferris wheel with a 50-foot radius. We will track the height and co-height of passenger cars that begin at the 3 o’clock position. Sketch graphs of the height and co-height functions for the following scenarios. a. A passenger car on the Ferris wheel completes one turn, traveling counterclockwise.

b. A passenger car on the Ferris wheel completes two full turns, traveling counterclockwise.


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c. The Ferris wheel is stuck in reverse, and a passenger car on the Ferris wheel completes two full clockwise turns.

4. Consider a Ferris wheel with radius of 40 feet that is rotating counterclockwise. At which amounts of rotation are the values of the height and co-height functions equal? Does this result hold for a Ferris wheel with a different radius?

5. Yuki is on a passenger car of a Ferris wheel at the 3 o’clock position. The wheel then rotates 135 degrees counterclockwise and gets stuck. Lee argues that she can compute the value of the co-height of Yuki’s car if she is given one of the following two pieces of information:

i. The value of the height function of Yuki’s car, or ii. The diameter of the Ferris wheel itself.

Is Lee correct? Explain how you know.


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SPIRAL REVIEW – Pythagorean Theorem For each right triangle below, find the missing side. Round your answer to the nearest tenth. [source: http://www.kutasoftware.com/FreeWorksheets/PreAlgWorksheets/Pythagorean%20Theorem.pdf]

6. 7.

8. 9.

10. 11.


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http://www.kutasoftware.com/FreeWorksheets/PreAlgWorksheets/Pythagorean%20Theorem.pdf


Lesson 4: The Unit Circle

Building the Unit Circle

1. The Unit Circle gets its name from the fact that its radius is 1 unit. Use that idea to write in the coordinates for the four points that lie on the axes. One has been done for you.

2. Let’s get the angle measure at each point on the Unit Circle. Use what you learned in Lessons 1 – 3 to write in each angle measure. Two have been done for you. (We will continually to go back to this page to fill in more and more of the circle.)

0° (1, 0)

30°

Lesson 4: The Unit Circle Unit 6: Trigonometry – Focus on Sine, Cosine and Tangent

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In Geometry, you studied special right triangles (30°-60°-90° and 45°-45°-90°). Those triangles play a major role in trigonometry and we’ll use them to get the remaining coordinates on the Unit Circle.

3. Let’s work with the 45°-45°-90° triangle. Divided the square below into two 45°-45°-90° triangles. If the hypotenuse of each triangle is 1, what are the side lengths? Label all the angles and segments with their measure.

4. The 30°-60°-90° triangle is derived from an equilateral triangle. Divided the equilateral triangle below into two 30°-60°-90° triangles. If the hypotenuse of each right triangle is 1, what are all the side lengths? Label all the angles and segments with their measure.

You will need: scissors, special triangles handout

5. Cut out the two special triangles on your handout and write the lengths and angle measures based on your work in Exercises 3 and 4. Write the measurements on both sides of the triangles. Then use them to determine the coordinates of each point on the unit circle on the previous page.


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The Unit Circle Revisited 6. Fill in each mini-unit circle below by writing in the degree measure and coordinates. Each one

focuses on one type of special angle. Try not to look back on your Unit Circle on page S.29, but you may use your triangles (from Exercise 5) to help you remember the coordinates.

Quadrantal Angles Angles that lie on the x- or y-axis

45° Angles

30° Angles 60° Angles


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Lesson Summary


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Homework Problem Set 1. Find the lengths of the sides of the right triangles below, each of which has hypotenuse of length 1.

2. Fill in the chart. Write in the coordinates for the indicated values of 𝜃𝜃.

Amount of rotation, 𝜽𝜽, in degrees

Coordinates

120

135

150

225

240

300

330


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3. Sketch each of the following angles in standard position (0° at the 3 o’clock position). Do not use a protractor. Estimate the location of each angle.

A. θ = 45° B. θ = 210° C. θ = 150°

D. θ = 180° E. θ = 60° F. θ = 300°

4. Determine the quadrant each angle is in. You do not have to draw each one.

[image source: https://www.wyzant.com/resources/lessons/math/trigonometry/unit-circle]

SPIRAL REVIEW - SohCahToa

Determine the sin A, cos A and tan A for each triangle. 5. 6. 7.

A. 135° B. 225° C. 92°

E. 307° F. 245° G. 335°


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https://www.wyzant.com/resources/lessons/math/trigonometry/unit-circle


Lesson 5: From Circle-ometry to Trigonometry

1. Suppose that point 𝑃𝑃 is the point on the unit circle obtained by rotating the initial ray through 30°.

A. What is the length 𝑂𝑂𝑂𝑂, the horizontal leg of our triangle? Write it on the unit circle above.

B. What is the length 𝑂𝑂𝑃𝑃,the vertical leg of our triangle? Write it on the unit circle above.

C. What is sin(30°)? D. What is cos(30°)?

2. Let’s compare the values you got in Exercise 1C and 1D with the coordinates on the unit circle.

A. What do you notice?

B. What can we say about the sine value of any angle on the unit circle?

C. What can we say about the cosine value of any angle on the unit circle?

The initial ray is the ray formed by the

positive x-axis. It is where the angle begins

its rotation.

The terminal ray defines the angle.

It is where the angle stops.

Lesson 5: From Circle-ometry to Trigonometry Unit 6: Trigonometry – Focus on Sine, Cosine and Tangent

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Use the unit circle to find each trigonometric value in the exercises below. 3. Suppose that 𝑃𝑃 is the point on the unit circle obtained by

rotating the initial ray through 45°. Find sin(45°) and cos(45°). 4. Suppose that 𝑃𝑃 is the point on the unit circle obtained by

rotating the initial ray through 60°. Find sin(60°) and cos(60°).

Discussion

Remember that sine and cosine are functions of the number of degrees of rotation of the initial horizontal ray moving counterclockwise about the origin. So far, we have only made sense of sine and cosine for degrees of rotation between 0 and 90, but the Ferris wheel doesn’t just rotate 90° and then stop; it continues going around the full circle. How can we extend our ideas about sine and cosine to any counterclockwise rotation up to 360°?

5. Suppose that 𝑃𝑃 is the point on the unit circle obtained by rotating the initial ray through 150°. How can we find sin(150°) and cos(150°)?

6. A. What are the coordinates (𝑥𝑥𝜃𝜃,𝑦𝑦𝜃𝜃) of point 𝑃𝑃? B. What is sin(150°)? C. What is cos(150°)?


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Reference Angles

In general if we rotate the initial ray through more than 90°, then the reference angle is the acute angle formed by the terminal ray and the 𝑥𝑥-axis. In the following diagrams, the measure of the reference angle is denoted by 𝜙𝜙, the Greek letter phi. Let’s start with the case where the terminal ray is rotated into the second quadrant.

If 90 < 𝜃𝜃 < 180, then the terminal ray of the rotation by 𝜃𝜃° lies in the second quadrant. The reference angle formed by the terminal ray and the 𝑥𝑥-axis has measure 𝜙𝜙° and is shaded in the figure on the right above. 7. A. How does 𝜙𝜙 relate to 𝜃𝜃?

B. How can we find the lengths 𝑂𝑂𝑂𝑂 and 𝑃𝑃𝑂𝑂?

C. How can we use these lengths to find the coordinates of point 𝑃𝑃?

8. If 90 < 𝜃𝜃 < 180, then rotation by 𝜃𝜃 degrees places 𝑃𝑃 in the second quadrant, with reference angle of measure 𝜙𝜙 degrees. Then what are the values of cos(𝜃𝜃°) and sin(𝜃𝜃°)?

cos(𝜃𝜃°) = _______________

sin(𝜃𝜃°) = _______________

9. What is cos(135°)? What is sin(135°)?


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In the diagram below, 180 < 𝜃𝜃 < 270, so that point 𝑃𝑃 is in the third quadrant.

10. What do we know about the 𝑥𝑥-coordinate and 𝑦𝑦-coordinate of 𝑃𝑃? If 180 < 𝜃𝜃 < 270, then the terminal ray of the rotation by 𝜃𝜃° lies in the third quadrant. The reference angle formed by the terminal ray and the 𝑥𝑥-axis has measure 𝜙𝜙° and is shaded in the figure on the right above. 11. A. How does 𝜙𝜙 relate to 𝜃𝜃?

B. How can we find the lengths 𝑂𝑂𝑂𝑂 and 𝑃𝑃𝑂𝑂? C. How can we use these lengths to find the coordinates of point 𝑃𝑃?

12. If 180 < 𝜃𝜃 < 270, then rotation by 𝜃𝜃 degrees places 𝑃𝑃 in the third quadrant, with reference angle of

measure 𝜙𝜙. Then what are the values of cos(𝜃𝜃°) and sin(𝜃𝜃°)?

cos(𝜃𝜃°) = ___________ sin(𝜃𝜃°) = ___________



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In the diagram below, 270 < 𝜃𝜃 < 360, so that point 𝑃𝑃 is in the fourth quadrant.

14. What do we know about the 𝑥𝑥-coordinate and 𝑦𝑦-coordinate of 𝑃𝑃?

If 270 < 𝜃𝜃 < 360, then the terminal ray of the rotation by 𝜃𝜃° lies in the fourth quadrant. The reference angle formed by the terminal ray and the 𝑥𝑥-axis has measure 𝜙𝜙° and is shaded in the figure on the right above. 15. A. How does 𝜙𝜙 relate to 𝜃𝜃?

B. How can we use the lengths 𝑂𝑂𝑂𝑂 and 𝑃𝑃𝑂𝑂 to find the coordinates of point 𝑃𝑃?

16. If 270 < 𝜃𝜃 < 360, then rotation by 𝜃𝜃 degrees places 𝑃𝑃 in the fourth quadrant, with reference angle of measure 𝜙𝜙 degrees. Then what are the values of cos(𝜃𝜃°) and sin(𝜃𝜃°)?

cos(𝜃𝜃°) = ___________ sin(𝜃𝜃°) = ___________



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What we have just concluded is very important. We have just extended the definitions of sine and cosine from Geometry to almost any number of degrees of rotation between 0 and 360, when they were previously only defined for 0 < 𝜃𝜃 < 90. Lesson 6 extends the domain of the sine and cosine even further by exploring what happens if 𝜃𝜃 > 360 and what happens if 𝜃𝜃 ≤ 0.

Discussion

18. How do you know whether cos(𝜃𝜃°) and sin(𝜃𝜃°) are positive or negative in each quadrant?

19. Suppose that 𝑃𝑃 is the point on the unit circle obtained by rotating the initial ray counterclockwise through 120 degrees. Find the measure of the reference angle for 120°, and then find sin(120°) and cos(120°).


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Lesson Summary

We have now made sense of the sine and cosine functions for nearly all values of theta with 0 < 𝜃𝜃 < 360, where 𝜃𝜃 is measured in degrees. In the next lesson, we extend the domains of these two functions even further, so that they are defined for any real number 𝜃𝜃.

The values of the sine and cosine functions at rotations of 30, 45, and 60 degrees and multiples of these rotations come up often in trigonometry. The diagram below summarizes the coordinates of these commonly referenced points.

In this lesson we formalized the idea of the height and co-height of a Ferris wheel and defined the sine and cosine functions that give the 𝑥𝑥- and 𝑦𝑦- coordinates of the intersection of the unit circle and the initial ray rotated through 𝜃𝜃 degrees, for most values of 𝜃𝜃 with 0 < 𝜃𝜃 < 360.

The value of cos(𝜃𝜃°) is the 𝑥𝑥-coordinate of the intersection point of the terminal ray and the unit circle.

The value of sin(𝜃𝜃°) is the 𝑦𝑦-coordinate of the intersection point of the terminal ray and the unit circle.

The sine and cosine functions have domain of all real numbers and range [−1,1]. The reference angle is the acute angle formed by the terminal ray and the 𝑥𝑥-axis.


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Homework Problem Set 1. Fill in the chart. Write in the reference angles and the values of the sine and cosine functions for the

indicated values of 𝜃𝜃.

Amount of rotation,

𝜽𝜽, in degrees

Measure of Reference Angle,

in degrees 𝐜𝐜𝐜𝐜𝐜𝐜(𝜽𝜽°) 𝐜𝐜𝐬𝐬𝐬𝐬(𝜽𝜽°)

120

135

150

225

240

300

330

2. Suppose that 𝑃𝑃 is the point on the unit circle obtained by rotating the initial ray counterclockwise through

240°. Find the measure of the reference angle for 240°, and then find sin(240°) and cos(240°). 3. Suppose that 𝑃𝑃 is the point on the unit circle obtained by rotating the initial ray counterclockwise through

330 degrees. Find the measure of the reference angle for 330°, and then find sin(330°) and cos(330°).


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4. Using geometry, Jennifer correctly calculated that sin(15°) = 12

�2 − √3 . Based on this information, fill in the chart.

Amount of rotation,

𝜽𝜽, in degrees

Measure of Reference Angle, in

degrees 𝐜𝐜𝐬𝐬𝐬𝐬(𝜽𝜽°)

15

165

195

345

For Problems 5 – 7, use the fact that in a unit circle x2 + y2 = 1, sinθ = y and cosθ = x.

5. Suppose 0 < 𝜃𝜃 < 90 and sin(𝜃𝜃°) = 1√3

. What is the value of cos(𝜃𝜃°)?

6. Suppose 90 < 𝜃𝜃 < 180 and sin(𝜃𝜃°) = 1√3

. What is the value of cos(𝜃𝜃°)?

7. If cos(𝜃𝜃°) = − 1√5

, what are two possible values of sin(𝜃𝜃°)?


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8. Johnny rotated the initial ray through 𝜃𝜃 degrees, found the intersection of the terminal ray with the unit circle, and calculated that sin(𝜃𝜃°) = √2. Ernesto insists that Johnny made a mistake in his calculation. Explain why Ernesto is correct.

9. If sin(𝜃𝜃°) = 0.5, and we know that cos (𝜃𝜃°) < 0, then what is the smallest possible positive value of 𝜃𝜃?

10. The vertices of triangle △ 𝐴𝐴𝐴𝐴𝐴𝐴 have coordinates 𝐴𝐴(0,0), 𝐴𝐴(12,5), and 𝐴𝐴(12,0). a. Argue that △ 𝐴𝐴𝐴𝐴𝐴𝐴 is a right triangle.

b. What are the coordinates where the hypotenuse of △ 𝐴𝐴𝐴𝐴𝐴𝐴 intersects the unit circle 𝑥𝑥2 + 𝑦𝑦2 = 1?

c. Let 𝜃𝜃 denote the number of degrees of rotation from 𝐴𝐴𝐴𝐴��⃗ to 𝐴𝐴𝐴𝐴��⃗ . Calculate sin(𝜃𝜃°) and cos(𝜃𝜃°).


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11. The vertices of triangle △ 𝐴𝐴𝐴𝐴𝐴𝐴 have coordinates 𝐴𝐴(0,0), 𝐴𝐴(4,3), and 𝐴𝐴(4,0). The vertices of triangle △𝐴𝐴𝐴𝐴𝐴𝐴 are 𝐴𝐴(0,0), 𝐴𝐴(3,4), and 𝐴𝐴(3,0). a. Argue that △ 𝐴𝐴𝐴𝐴𝐴𝐴 is a right triangle.

b. What are the coordinates where the hypotenuse of △ 𝐴𝐴𝐴𝐴𝐴𝐴 intersects the unit circle 𝑥𝑥2 + 𝑦𝑦2 = 1?

c. Let 𝜃𝜃 denote the number of degrees of rotation from 𝐴𝐴𝐴𝐴��⃗ to 𝐴𝐴𝐴𝐴��⃗ . Calculate sin(𝜃𝜃°) and cos(𝜃𝜃°).

d. Argue that △ 𝐴𝐴𝐴𝐴𝐴𝐴 is a right triangle.

e. What are the coordinates where the hypotenuse of △ 𝐴𝐴𝐴𝐴𝐴𝐴 intersects the unit circle 𝑥𝑥2 + 𝑦𝑦2 = 1?

f. Let 𝜙𝜙 denote the number of degrees of rotation from 𝐴𝐴𝐴𝐴��⃗ to 𝐴𝐴𝐴𝐴��⃗ . Calculate sin(𝜙𝜙°) and cos(𝜙𝜙°).

g. What is the relation between the sine and cosine of 𝜃𝜃 and the sine and cosine of 𝜙𝜙?


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12. Use a diagram to explain why sin(135°) = sin(45°), but cos(135°) ≠ cos(45°).


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Lesson 6: Extending the Domain of Sine and Cosine to All Real Numbers

Opening Exercise

In Lesson 5, we found a way to calculate the sine and cosine functions for rotations of the initial ray (made from the positive 𝑥𝑥-axis) through 𝜃𝜃 degrees, for 0 < 𝜃𝜃 < 360. Today, we investigate what happens if 𝜃𝜃 takes on a value outside of the interval (0, 360). Remember that our motivating examples for the sine and cosine functions were the height and co-height functions associated with a rotating Ferris wheel. Let’s return to that context for this discussion. 1. In reality, a Ferris wheel doesn’t just go around once and then stop. It rotates a number of times and

then stops to let the riders off. How can we extend our ideas about sine and cosine to a counterclockwise rotation through more than 360°?

ANGLES GREATER THAN 360°

Suppose that 𝑃𝑃 is the point on the unit circle obtained from rotating the initial ray through 390° and we want to find sin(390°) and cos(390°).

2. Does it make sense to think of a reference angle for this rotation? Explain your thinking.

3. What is the measure of the reference angle for this rotation?

4. What are the coordinates (𝑥𝑥𝜃𝜃,𝑦𝑦𝜃𝜃) of point 𝑃𝑃?

5. What is sin(390°)? What is cos (390°)?

6. Find cos(405°) and sin(405°). Identify the measure of the reference angle.

Lesson 6: Extending the Domain of Sine and Cosine to All Real Numbers Unit 6: Trigonometry – Focus on Sine, Cosine and Tangent

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10. Find cos(720 030°) and sin(720 030°). Identify the measure of the reference angle.

ANGLES LESS THAN 0° Now we know how to calculate the values of the sine and cosine functions for rotating further than 360° counterclockwise. But what if the Ferris wheel malfunctions and starts rotating backward? Does it still make sense to talk about the height and co-height functions if the Ferris wheel is turning the wrong way?

11. In our definition of sine and cosine, how can we indicate that the rotation is happening in the opposite

direction from our normal counterclockwise rotation?


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12. Suppose that 𝑃𝑃 is the point on the unit circle obtained from rotating the initial ray through −150°. We want to find sin (−150°) and cos(−150°).

A. What is the measure of the reference angle for ∠𝑃𝑃𝑃𝑃𝑃𝑃?

B. What are the coordinates (𝑥𝑥𝜃𝜃,𝑦𝑦𝜃𝜃) of point 𝑃𝑃?

C. What is sin(−150°)? D. What is cos(−150°)?

13. Find cos(−30°) and sin(−30°). Identify the measure of the reference angle.




S.49










Discussion

At this point, we have defined the sine and cosine functions for almost any positive or negative rotation, but there are a few cases we have not yet addressed. What if the Ferris wheel completely breaks down and will not move at all once you have been loaded into your car? Does it still make sense to talk about the height and co-height functions if the Ferris wheel never gets started? Can we still think of the car as rotating through a number of degrees?

Case 1: What about the values of the sine and cosine function of other amounts of rotation that produce a terminal ray along the positive 𝑥𝑥-axis, such as 1080°?

Our definition of a reference angle is the angle formed by the terminal ray and the 𝑥𝑥-axis, but our terminal ray lies along the 𝑥𝑥-axis so the terminal ray and the 𝑥𝑥-axis form a zero angle.

18. How would we assign values to cos(1080°) and sin(1080°)?

19. What if we rotated around 24,000°, which is 400 turns? What are cos (24000°) and sin(24000°)?

20. State a generalization of these results:

If 𝜃𝜃 = 𝑛𝑛 ∙ 360, for some integer 𝑛𝑛, then cos(𝜃𝜃°) = _____, and sin(𝜃𝜃°) = ______.


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Case 2: What about the values of the sine and cosine function of other amounts of rotation that produce a terminal ray along the negative 𝑥𝑥-axis, such as 540°?

21. How would we assign values to cos(540°) and sin(540°)?

22. What are the values of cos (900°) and sin(900°)? How do you know?


If 𝜃𝜃 = 𝑛𝑛 ∙ 360 + 180, for some integer 𝑛𝑛, then cos(𝜃𝜃°) = _____, and sin(𝜃𝜃°) = ______.

Case 3: What about the values of the sine and cosine function for rotations that are 90° more than a number of full turns, such as −630°? How would we assign values to cos(−630°), and sin(−630°)?

24. Can we generalize to any rotation that produces a terminal ray along the positive 𝑦𝑦-axis?




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Case 4: What about the values of the sine and cosine function for rotations whose terminal ray lies along the negative 𝑦𝑦-axis, such as −810°?

26. How would we assign values to cos(−810°) and sin(−810°)?

27. Can we generalize to any rotation that produces a terminal ray along the negative 𝑦𝑦-axis?



Lesson Summary

In this lesson the definition of the sine and cosine are formalized as functions of a number of degrees of rotation, 𝜃𝜃. The initial ray made from the positive 𝑥𝑥-axis through 𝜃𝜃 degrees is rotated, going counterclockwise if 𝜃𝜃 > 0 and clockwise if 𝜃𝜃 < 0. The point 𝑃𝑃 is defined by the intersection of the terminal ray and the unit circle.

The value of cos(𝜃𝜃°) is the 𝑥𝑥-coordinate of 𝑃𝑃. The value of sin(𝜃𝜃°) is the 𝑦𝑦-coordinate of 𝑃𝑃. The sine and cosine functions have domain of all real numbers and range [−1,1].


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Homework Problem Set 1. Fill in the chart. Write in the measures of the reference angles and the values of the sine and cosine

functions for the indicated values of 𝜃𝜃.

Number of degrees of rotation,

𝜽𝜽 Quadrant

Measure of Reference

Angle, in degrees 𝐜𝐜𝐜𝐜𝐜𝐜(𝜽𝜽°) 𝐜𝐜𝐬𝐬𝐬𝐬(𝜽𝜽°)

690

810

1560

1440

855

−330

−4500

−510

−135

−1170

2. Using geometry, Jennifer correctly calculated that sin(15°) = 12�2 −√3. Based on this information, fill

in the chart:

Number of degrees of rotation,

𝜽𝜽 Quadrant

Measure of Reference Angle,

in degrees 𝐜𝐜𝐜𝐜𝐜𝐜(𝜽𝜽°) 𝐜𝐜𝐬𝐬𝐬𝐬(𝜽𝜽°)

525

705

915

−15

−165

−705


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3. Suppose 𝜃𝜃 represents a number of degrees of rotation and that sin(𝜃𝜃°) = 0.5. List the first six possible positive values that 𝜃𝜃 can take.

4. Suppose 𝜃𝜃 represents a number of degrees of rotation and that sin(𝜃𝜃°) = −0.5. List six possible negative values that 𝜃𝜃 can take.

5. Suppose 𝜃𝜃 represents a number of degrees of rotation. Is it possible that cos(𝜃𝜃°) = 12 and sin(𝜃𝜃°) = 1

2?

6. Jane says that since the reference angle for a rotation through −765° has measure 45°, then cos(−765°) = cos(45°), and sin(−765°) = sin(45°). Explain why she is or is not correct.

7. Doug says that since the reference angle for a rotation through 765° has measure 45°, then cos(765°) =cos(45°), and sin(765°) = sin(45°). Explain why he is or is not correct.


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Lesson 7: Why Call It Tangent?

Opening Exercise

1. Let 𝑃𝑃(𝑥𝑥𝜃𝜃,𝑦𝑦𝜃𝜃) be the point where the terminal ray intersects the unit circle after rotation by 𝜃𝜃 degrees, as shown in the diagram below.

A. Using triangle trigonometry, what are the values of 𝑥𝑥𝜃𝜃 and 𝑦𝑦𝜃𝜃 in terms of 𝜃𝜃?

B. Using triangle trigonometry, what is the value of tan(𝜃𝜃°) in terms of 𝑥𝑥𝜃𝜃 and 𝑦𝑦𝜃𝜃?

C. What is the value of tan(𝜃𝜃°) in terms of 𝜃𝜃?

D. Which θ values will give the tan (θ°) as undefined?

Lesson 7: Why Call It Tangent? Unit 6: Trigonometry – Focus on Sine, Cosine and Tangent

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2. For each value of 𝜃𝜃 in the table below, use the given values of sin(𝜃𝜃°) and cos(𝜃𝜃°) to approximate tan(𝜃𝜃°) to two decimal places.

𝜽𝜽 (degrees) 𝐬𝐬𝐬𝐬𝐬𝐬(𝜽𝜽°) 𝐜𝐜𝐜𝐜𝐬𝐬(𝜽𝜽°) 𝐭𝐭𝐭𝐭𝐬𝐬(𝜽𝜽°)

−89.9 −0.999998 0.00175

−89 −0.9998 0.0175

−85 −0.996 0.087

−80 −0.98 0.17

−60 −0.87 0.50

−40 −0.64 0.77

−20 −0.34 0.94

0 0 1.00

20 0.34 0.94

40 0.64 0.77

60 0.87 0.50

80 0.98 0.17

85 0.996 0.087

89 0.9998 0.0175

89.9 0.999998 0.00175

Discussion 3. A. As 𝜃𝜃 → −90° and 𝜃𝜃 > −90°, what value does sin(𝜃𝜃°) approach?

B. As 𝜃𝜃 → −90° and 𝜃𝜃 > −90°, what value does cos(𝜃𝜃°) approach?

C. As 𝜃𝜃 → −90° and 𝜃𝜃 > −90°, how would you describe the value of tan(𝜃𝜃°) = sin(𝜃𝜃°)cos(𝜃𝜃°)?


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4. A. As 𝜃𝜃 → 90° and 𝜃𝜃 < 90°, what value does sin(𝜃𝜃°) approach?

B. As 𝜃𝜃 → 90° and 𝜃𝜃 < 90°, what value does cos(𝜃𝜃°) approach?

C. As 𝜃𝜃 → 90° and 𝜃𝜃 < 90°, how would you describe the behavior of tan(𝜃𝜃°) = sin(𝜃𝜃°)cos(𝜃𝜃°)?

5. How can we describe the range of the tangent function?

6. Suppose that point 𝑃𝑃 is the point on the unit circle obtained by rotating the initial ray through 30°. Find tan(30°).


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Discussion – Why Do We Call it Tangent?

A description of the tangent function is provided below. Be prepared to answer questions based on your understanding of this function and to discuss your responses with others in your class.

7. Let 𝑃𝑃 be the point on the unit circle with center 𝑂𝑂 that is the intersection of the terminal ray after rotation by 𝜃𝜃 degrees as shown in the diagram. Let 𝑄𝑄 be the foot of the perpendicular line from 𝑃𝑃 to the 𝑥𝑥-axis, and let the line ℓ be the line perpendicular to the 𝑥𝑥-axis at 𝑆𝑆(1,0). Let 𝑅𝑅 be the point where the secant line 𝑂𝑂𝑃𝑃 intersects the line ℓ. Let 𝑚𝑚 be the length of 𝑅𝑅𝑆𝑆��.

A. Show that 𝑚𝑚 = tan(𝜃𝜃°).

B. Using a segment in the figure, make a conjecture why mathematicians named the function 𝑓𝑓(𝜃𝜃°) = sin(𝜃𝜃°)cos(𝜃𝜃°)

the tangent function.

C. Why can you use either triangle, △ 𝑃𝑃𝑂𝑂𝑄𝑄 or △ 𝑅𝑅𝑂𝑂𝑆𝑆, to calculate the length 𝑚𝑚?

D. Imagine that you are the mathematician who gets to name the function. (How cool would that be?) Based upon what you know about the equations of lines, what might you have named the function instead?


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8. A. Draw four pictures similar to the diagram in Exercise 7 to illustrate what happens to the value of tan(𝜃𝜃°) as the rotation of the secant line through the terminal ray increases towards 90°. How does your diagram relate to the work done in Exercise 2?

C. When the terminal ray is vertical, what is the relationship between the secant line 𝑂𝑂𝑅𝑅 and the tangent

line 𝑅𝑅𝑆𝑆? Explain why you cannot determine the measure of 𝑚𝑚 in this instance. What is the value of tan(90°)?

D. When the terminal ray is horizontal, what is the relationship between the secant line 𝑂𝑂𝑅𝑅 and the 𝑥𝑥-

axis? Explain what happens to the value of 𝑚𝑚 in this instance. What is the value of tan(0°)?

E. When the terminal ray is rotated counterclockwise about the origin by 45°, what is the relationship between the value of 𝑚𝑚 and the length of 𝑂𝑂𝑆𝑆��? What is the value of tan(45°)?


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9. Rotate the initial ray about the origin the stated number of degrees. Draw a sketch and label the coordinates of point 𝑃𝑃 where the terminal ray intersects the unit circle. What is the slope of the line containing this ray? A. 30°

B. 𝟒𝟒𝟒𝟒°

C. 60°

D. Use the definition of tangent to find tan (30°), tan (45°), and tan (60°). How do your answers compare your work in Parts A – C?


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10. If the initial ray is rotated 𝜃𝜃 degrees about the origin, show that the slope of the line containing the

terminal ray is equal to tan(𝜃𝜃°). Explain your reasoning.

11. Now that you have shown that the value of the tangent function is equal to the slope of the terminal ray, would you prefer using the name tangent function or slope function? Why do you think we use tangent instead of slope as the name of the tangent function?

12. Rotate the initial ray about the origin the stated number of degrees. Draw a sketch and label the coordinates of point 𝑃𝑃 where the terminal ray intersects the unit circle. How does your diagram in this exercise relate to the diagram in the corresponding part of Exercise 9? What is tan(𝜃𝜃°) for these values of 𝜃𝜃? A. 210° B. 225° C. 240°

D. What do the results of Parts A – C suggest about the value of the tangent function after rotating an additional 180 degrees?

E. What is the period of the tangent function? Discuss with a classmate and write your conclusions.


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F. Use the results of Exercise 10 to explain why tan(0°) = 0.

G. Use the results of Exercise 10 to explain why tan(90°) is undefined. 13. Consider the diagram of a circle of radius 𝑟𝑟 centered at the origin. The line ℓ is tangent to the circle at

𝑆𝑆(𝑟𝑟, 0), so ℓ is perpendicular to the 𝑥𝑥-axis.

A. If 𝑟𝑟 = 1, then state the value of 𝑡𝑡 in terms of one of the trigonometric functions.

B. If 𝑟𝑟 is any positive value, then state the value of 𝑡𝑡 in terms

of one of the trigonometric functions.

Lesson Summary

A working definition of the tangent function is tan(𝜃𝜃°) = sin(𝜃𝜃°)cos(𝜃𝜃°), where cos(𝜃𝜃°) ≠ 0.

The value of tan(𝜃𝜃°) is the length of the line segment on the tangent line to the unit circle centered at the origin from the intersection with the unit circle and the intersection with the secant line created by the 𝑥𝑥-axis rotated 𝜃𝜃 degrees. (This is why we call it tangent.)

The value of tan(𝜃𝜃°) is the slope of the line obtained by rotating the 𝑥𝑥-axis 𝜃𝜃 degrees about the origin.

The domain of the tangent function is {𝜃𝜃 ∈ ℝ|𝜃𝜃 ≠ 90 + 180𝑘𝑘, for all integers 𝑘𝑘} which is equivalent to {𝜃𝜃 ∈ ℝ| cos(𝜃𝜃°) ≠ 0}.

The range of the tangent function is all real numbers. The period of the tangent function is 180°.

𝐭𝐭𝐭𝐭𝐬𝐬(𝟎𝟎°) 𝐭𝐭𝐭𝐭𝐬𝐬(𝟑𝟑𝟎𝟎°) 𝐭𝐭𝐭𝐭𝐬𝐬(𝟒𝟒𝟒𝟒°) 𝐭𝐭𝐭𝐭𝐬𝐬(𝟔𝟔𝟎𝟎°) 𝐭𝐭𝐭𝐭𝐬𝐬(𝟗𝟗𝟎𝟎°)

0 √33

1 √3 undefined


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Homework Problem Set Label the missing side lengths, and find the value of 𝐭𝐭𝐭𝐭𝐬𝐬(𝜽𝜽°) in the following right triangles. 1. 𝜃𝜃 = 30

2. 𝜃𝜃 = 45

3. 𝜃𝜃 = 60

30°

1

45°

1

60°

1


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4. Let 𝜃𝜃 be any real number. In the Cartesian plane, rotate the initial ray by 𝜃𝜃 degrees about the origin. Intersect the resulting terminal ray with the unit circle to get point 𝑃𝑃(𝑥𝑥𝜃𝜃,𝑦𝑦𝜃𝜃).

A. Complete the table by finding the slope of the line through the origin and the point 𝑃𝑃.

𝜽𝜽, in degrees Slope 𝜽𝜽, in degrees Slope

0 180

30 210

45 225

60 240

90 270

120 300

135 315

150 330

B. Explain how these slopes are related to the tangent function.

5. For the given values of 𝑟𝑟 and 𝜃𝜃, find 𝑡𝑡. A. 𝜃𝜃 = 30, 𝑟𝑟 = 2

B. 𝜃𝜃 = 45, 𝑟𝑟 = 2

C. 𝜃𝜃 = 60, 𝑟𝑟 = 2

D. 𝜃𝜃 = 45, 𝑟𝑟 = 4

E. 𝜃𝜃 = 30, 𝑟𝑟 = 3.5


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F. 𝜃𝜃 = 0, 𝑟𝑟 = 9

G. 𝜃𝜃 = 90, 𝑟𝑟 = 5

H. 𝜃𝜃 = 60, 𝑟𝑟 = √3

I. 𝜃𝜃 = 30, 𝑟𝑟 = 2.1

J. 𝜃𝜃 = 𝐴𝐴, 𝑟𝑟 = 3

K. 𝜃𝜃 = 30, 𝑟𝑟 = 𝑏𝑏

L. Knowing that tan(𝜃𝜃°) = sin(𝜃𝜃°)cos(𝜃𝜃°), for 𝑟𝑟 = 1, find the value of 𝑠𝑠 in terms of one of the trigonometric

functions.

M. Using what you know of the tangent function, show that −tan(𝜃𝜃°) = tan(−𝜃𝜃°) for 𝜃𝜃 ≠ 90 + 180𝑘𝑘, for all integers 𝑘𝑘.


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Lesson 8: Secant and the Co-Functions

Opening Exercise

The geometry of the unit circle and its related triangles provide a clue as to how the different reciprocal functions got their names. This lesson draws out the connections among tangent and secant lines of a circle, angle relationships, and the trigonometric functions. The names for the various trigonometric functions may make more sense when viewed through the lens of geometric figures.

1. Find the length of each segment below in terms of the value of a trigonometric function.

𝑂𝑂𝑂𝑂 = 𝑃𝑃𝑂𝑂 = 𝑅𝑅𝑅𝑅 =

Since there are ways to calculate lengths for nearly every line segment in this diagram using the length of the radius, or the cosine, sine, or tangent functions, it makes sense to find the length of 𝑂𝑂𝑅𝑅��, the line segment on the terminal ray that intersects the tangent line.

2. What do you call a line that intersects a circle at more than one point?

Lesson 8: Secant and the Co-Functions Unit 6: Trigonometry – Focus on Sine, Cosine and Tangent

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In Lesson 7, we saw that 𝑅𝑅𝑅𝑅 = tan(𝜃𝜃°), where 𝑅𝑅𝑅𝑅�� lies on the line tangent to the unit circle at (1,0), which helped to explain how this trigonometric function got its name. Let’s introduce a new function sec(𝜃𝜃°), the secant of 𝜃𝜃, to be the length of 𝑂𝑂𝑅𝑅�� since this segment is on the secant line that contains the terminal ray. Then the secant of 𝜃𝜃 is sec(𝜃𝜃°) = 𝑂𝑂𝑅𝑅.

3. We’ll use similar triangles to find the value of sec(𝜃𝜃°) in terms of one other trigonometric function.

A. Why are triangles POQ and ROS similar?

B. Redraw the triangles in the space below. Then highlight the sides that correspond to each other. (For example, use a yellow highlighter to mark

sides OP and OR since they are corresponding sides.)

C. Write a proportion using sides OP and OR .

D. Since the radius of the circle is 1, use this to simplify your proportion in Part C.

E. Use trigonometric expressions to rewrite your equation. (Remember you are trying to solve for OR .)


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4. The definition of the secant function is given below. Answer the questions to better understand this definition and the domain and range of this function. Be prepared to discuss your responses with others in your class.

A. What is the domain of the secant function?

B. The domains of the secant and tangent functions are the same. Why?

C. What is the range of the secant function? How is this range related to the range of the cosine function?

D. Is the secant function a periodic function? If so, what is its period?

• Let 𝜃𝜃 be any real number. • In the Cartesian plane, rotate the

nonnegative 𝑥𝑥-axis by 𝜃𝜃 degrees about the origin.

• Intersect this new ray with the unit circle to get a point (𝑥𝑥𝜃𝜃,𝑦𝑦𝜃𝜃).

• If 𝑥𝑥𝜃𝜃 ≠ 0, then the value of sec(𝜃𝜃°) is 1𝑥𝑥𝜃𝜃

.

• Otherwise, sec(𝜃𝜃°) is undefined. • In terms of the cosine function,

sec(𝜃𝜃°) = 1cos(𝜃𝜃°)

for cos(𝜃𝜃°) ≠ 0.


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Discussion

5. What are the values of sec(𝜃𝜃°) when the terminal ray is horizontal? When the terminal ray is vertical? 6. Name several values of θ for which sec(θ°) is undefined. Explain your reasoning. 7. How do the values of the secant and cosine functions vary with each other? As cos(𝜃𝜃°) gets larger, what

happens to the value of sec(𝜃𝜃°)? As cos(𝜃𝜃°) gets smaller but stays positive, what happens to the value of sec(𝜃𝜃°)? What about when cos(𝜃𝜃°) < 0?

8. What is the smallest positive value of sec(θ°)? Where does this occur? 9. What is the largest negative value of sec(𝜃𝜃°)? Where does this occur?

10. In the diagram, the horizontal line is tangent to the unit circle at (0,1).

a. How does this diagram compare to the one given in the Opening Exercise?

b. What is the relationship between 𝛽𝛽 and 𝜃𝜃?

c. Which segment in the figure has length sin(𝜃𝜃°)? Which segment has length cos(𝜃𝜃°)?


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d. Which segment in the figure has length sin(𝛽𝛽°)? Which segment has length cos(𝛽𝛽°)?

e. How can you write sin(𝜃𝜃°) and cos(𝜃𝜃°) in terms of the trigonometric functions of 𝛽𝛽?

11. The horizontal line is tangent to the circle at (0,1).

a. If two angles are complements with measures 𝛽𝛽 and 𝜃𝜃 as shown in the diagram, use similar triangles to show that sec(𝛽𝛽°) = 1

sin(𝜃𝜃°) .

b. If two angles are complements with measures 𝛽𝛽 and 𝜃𝜃 as shown in the diagram, use similar triangles to show that tan(𝛽𝛽°) = 1

tan(𝜃𝜃°).


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Discussion

Definitions of the cosecant and cotangent functions are offered below. Answer the questions to better understand the definitions and the domains and ranges of these functions. Be prepared to discuss your responses with others in your class.

12. The secant, cosecant, and cotangent functions are often referred to as reciprocal functions. Why do you think these functions are so named?

13. Why are the domains of these functions restricted?

14. The domains of the cosecant and cotangent functions are the same. Why?

15. What is the range of the cosecant function? How is this range related to the range of the sine function?

16. What is the range of the cotangent function? How is this range related to the range of the tangent function?

• Let 𝜃𝜃 be any real number such that 𝜃𝜃 ≠ 180𝑘𝑘 for all integers 𝑘𝑘.

• In the Cartesian plane, rotate the initial ray by 𝜃𝜃 degrees about the origin. Intersect the resulting terminal ray with the unit circle to get a point (𝑥𝑥𝜃𝜃,𝑦𝑦𝜃𝜃).

• The value of csc(𝜃𝜃°) is 1𝑦𝑦𝜃𝜃

.

• The value of cot(𝜃𝜃°) is 𝑥𝑥𝜃𝜃𝑦𝑦𝜃𝜃

.


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Lesson Summary

Let 𝜃𝜃 be any real number. In the Cartesian plane, rotate the initial ray by 𝜃𝜃 degrees about the origin. Intersect the resulting terminal ray with the unit circle to get a point (𝑥𝑥𝜃𝜃,𝑦𝑦𝜃𝜃). Then:

Function Value For any 𝜽𝜽 such that… Formula

Sine 𝑦𝑦𝜃𝜃 𝜃𝜃 is a real number

Cosine 𝑥𝑥𝜃𝜃 𝜃𝜃 is a real number

Tangent 𝑦𝑦𝜃𝜃𝑥𝑥𝜃𝜃

𝜃𝜃 ≠ 90 + 180𝑘𝑘, for all integers 𝑘𝑘 tan(𝜃𝜃°) =sin(𝜃𝜃°)cos(𝜃𝜃°)

Secant 1𝑥𝑥𝜃𝜃

𝜃𝜃 ≠ 90 + 180𝑘𝑘, for all integers 𝑘𝑘 sec(𝜃𝜃°) =1

cos(𝜃𝜃°)

Cosecant 1𝑦𝑦𝜃𝜃

𝜃𝜃 ≠ 180𝑘𝑘, for all integers 𝑘𝑘 csc(𝜃𝜃°) =1

sin(𝜃𝜃°)

Cotangent 𝑥𝑥𝜃𝜃𝑦𝑦𝜃𝜃

𝜃𝜃 ≠ 180𝑘𝑘, for all integers 𝑘𝑘 cot(𝜃𝜃°) =cos(𝜃𝜃°)sin(𝜃𝜃°)


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Homework Problem Set 1. Use the reciprocal interpretations of sec(𝜃𝜃°), csc(𝜃𝜃°), and

cot(𝜃𝜃°) and the unit circle provided to complete the table.

𝜽𝜽, in degrees 𝐬𝐬𝐬𝐬𝐬𝐬(𝜽𝜽°) 𝐬𝐬𝐬𝐬𝐬𝐬(𝜽𝜽°) 𝐬𝐬𝐜𝐜𝐜𝐜(𝜽𝜽°)

0

30

45

60

90

120

180

225

240

270

315

330


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2. Find the following values from the information given. a. sec(𝜃𝜃°); cos(𝜃𝜃°) = 0.3

b. csc(𝜃𝜃°); sin(𝜃𝜃°) = −0.05

c. cot(𝜃𝜃°); tan(𝜃𝜃°) = 1000

d. sec(𝜃𝜃°); cos(𝜃𝜃°) = −0.9

e. csc(𝜃𝜃°); sin(𝜃𝜃°) = 0

f. cot(𝜃𝜃°); tan(𝜃𝜃°) = −0.0005

3. Choose three 𝜃𝜃 values from the table in Problem 1 for which sec(𝜃𝜃°), csc(𝜃𝜃°), and tan(𝜃𝜃°) are defined

and not zero. Show that for these values of 𝜃𝜃, sec(𝜃𝜃°)csc(𝜃𝜃°) = tan(𝜃𝜃°).


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4. Find the value of sec(𝜃𝜃°)cos(𝜃𝜃°) for the following values of 𝜃𝜃. a. 𝜃𝜃 = 120

b. 𝜃𝜃 = 225

c. 𝜃𝜃 = 330

d. Explain the reasons for the pattern you see in your responses to parts (a)–(c).

5. Draw a diagram representing the two values of 𝜃𝜃 between 0 and 360 so that sin(𝜃𝜃°) = −√32

. Find the values of tan(𝜃𝜃°), sec(𝜃𝜃°), and csc(𝜃𝜃°) for each value of 𝜃𝜃.

6. Find the value of �sec(𝜃𝜃°)�2 − �tan(𝜃𝜃°)�2 when 𝜃𝜃 = 225.

7. Find the value of �csc(𝜃𝜃°)�2 − �cot(𝜃𝜃°)�2 when 𝜃𝜃 = 330.


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CHALLENGE PROBLEMS

8. Using the formulas sec(𝜃𝜃°) = 1cos(θ°), csc(𝜃𝜃°) = 1

sin(𝜃𝜃°), and cot(𝜃𝜃°) = 1tan(𝜃𝜃°), show that sec(𝜃𝜃°)

csc(𝜃𝜃°) = tan(𝜃𝜃°),

where these functions are defined and not zero.

9. Tara showed that sec(𝜃𝜃°)csc(𝜃𝜃°) = tan(𝜃𝜃°), for values of 𝜃𝜃 for which the functions are defined and csc(𝜃𝜃°) ≠ 0,

and then concluded that sec(𝜃𝜃°) = sin(𝜃𝜃°) and csc(𝜃𝜃°) = cos(𝜃𝜃°). Explain what is wrong with her reasoning.


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10. From Lesson 7, Ren remembered that the tangent function is odd, meaning that −tan(𝜃𝜃°) = tan(−𝜃𝜃°) for all 𝜃𝜃 in the domain of the tangent function. He concluded because of the relationship between the secant function, cosecant function, and tangent function developed in Problem 9, it is impossible for both the secant and the cosecant functions to be odd. Explain why he is correct.


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Lesson 9: Graphing the Sine and Cosine Functions – Spaghetti Graphs

Exploratory Challenge

1. Your group will be graphing either: 𝑓𝑓(𝜃𝜃) = sin(𝜃𝜃°) or 𝑔𝑔(𝜃𝜃) = cos(𝜃𝜃°)

Your group will need:

• Blank unit circle

• Light-colored yarn – at least 30 inches

• Marker

• Spaghetti (uncooked, roughly 30 strands per group)

• 2 pieces of cardstock paper or poster board (that measures at least 9" × 30")

• Glue stick and tape

The circle on the next page is a unit circle, meaning that the length of the radius is one unit.

A. Mark axes on the poster board, with a horizontal axis in the middle of the board and a vertical axis near the left edge, as shown.

B. Measure the radius of the circle using a ruler. Use the length of the radius to mark 1 and −1 on the vertical axis.

C. Wrap the yarn around the circumference of the circle starting at 0. Mark each 15° increment on the yarn with the marker. Unwind the yarn and lay it on the horizontal axis. Transfer the marks on the yarn to corresponding increments on the horizontal axis. Label these marks as 0, 15, 30, …, 360.

D. Record the number of degrees of rotation 𝜃𝜃 on the horizontal axis of the graph, and record the value of either sin(𝜃𝜃°) or cos(𝜃𝜃°) on the vertical axis. Notice that the scale is wildly different on the vertical and horizontal axes.

E. If you are graphing 𝑔𝑔(𝜃𝜃) = cos(𝜃𝜃°): For each 𝜃𝜃 marked on your horizontal axis, beginning at 0, use the spaghetti to measure the horizontal displacement from the vertical axis to the relevant point on the unit circle. The horizontal displacement is the value of the cosine function. Break the spaghetti to mark the correct length, and place it vertically at the appropriate tick mark on the horizontal axis.

Lesson 9: Graphing the Sine and Cosine Functions – Spaghetti Graphs Unit 6: Trigonometry – Focus on Sine, Cosine and Tangent

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F. If you are graphing 𝑓𝑓(𝜃𝜃) = sin(𝜃𝜃°): For each 𝜃𝜃 marked on your horizontal axis, beginning at 0, use the spaghetti to measure the vertical displacement from the horizontal to the relevant point on the unit circle. The vertical displacement is the value of the sine function. Break the spaghetti to mark the correct length, and place it vertically at the appropriate tick mark on the horizontal axis.

G. Remember to place the spaghetti below the horizontal axis when the value of the sine function or the cosine function is negative. Glue each piece of spaghetti in place.

H. Draw a smooth curve that connects the points at the end of each piece of spaghetti.

Discussion

2. How are the graphs of the sine and cosine functions alike? 3. Could I get the graph of the sine function by shifting the graph of the cosine function? 4. If we extended the horizontal axis, what would the graph of the sine function look like between 360 and

720? What about from 720 to 1080? 5. What if we extended the graph in the negative direction?

6. At which values of θ is the sine function at a maximum? How about the cosine function?

7. At which values of θ does the sine function cross the x-axis? How about the cosine function?


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Lesson Summary

[source: http://www.electronics-tutorials.ws/accircuits/sinusoidal-waveform.html]

Homework Problem Set I. Write a paragraph comparing and contrasting the sine and cosine functions using their graphs and end

behavior.


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http://www.electronics-tutorials.ws/accircuits/sinusoidal-waveform.html


Use the graph of the sine function given below to answer Questions 2 - 4.

2. Desmond is trying to determine the value of sin(45°). He decides that since 45 is halfway between 0 and

90 that sin(45°) = 12. Use the graph to show him that he is incorrect.

3. Using the graph, complete each statement by filling in the symbol >, <, or =.

i. sin(250°) sin(290°)

ii. sin(25°) sin(85°)

iii. sin(140°) sin(160°)

On the interval [0,450], list the values of 𝜃𝜃 such that sin(𝜃𝜃°) = 12.

4. Explain why there are no values of 𝜃𝜃 such that sin(𝜃𝜃°) = 2.


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Lesson 10: Graphing the Sine and Cosine Functions – Looking for Patterns

Opening Exercise

1. Mark on the cartoon where the sine and cosine function are repeating. How many periods are shown on each graph?

Lesson 10: Graphing the Sine and Cosine Functions – Looking for Patterns Unit 6: Trigonometry – Focus on Sine, Cosine and Tangent

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Exploratory Challenge

2. Part I: Consider the function 𝑓𝑓(𝜃𝜃) = sin(𝜃𝜃°).

A. Complete the following table by using the special values learned in Lessons 4 and 5. Give values as approximations to one decimal place.

𝜃𝜃, in degrees 0 30 45 60 90 120 135 150 180

sin(𝜃𝜃°)

𝜃𝜃, in degrees 210 225 240 270 300 315 330 360

sin(𝜃𝜃°)

𝜃𝜃, in degrees 390 405 420 450 480 495 510 540

sin(𝜃𝜃°)

B. Using the values in the table, sketch the graph of the sine function on the interval [0, 540].


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3. A. For the interval [0, 540], describe the values of 𝜃𝜃 at which the sine function has relative maxima and minima.

B. For the interval l [0, 540], describe the values of 𝜃𝜃 for which the sine function is increasing and decreasing.

C. For the interval [0, 540], list the values of 𝜃𝜃 at which the graph of the sine function crosses the

horizontal axis.

D. Describe the end behavior of the sine function.

E. Based on the graph, is sine an odd function, even function, or neither? How do you know?

F. Describe how the sine function repeats.


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4. Part II: Consider the function 𝑔𝑔(𝜃𝜃) = cos(𝜃𝜃°).

A. Complete the following table giving answers as approximations to one decimal place.

𝜃𝜃, in degrees 0 30 45 60 90 120 135 150 180

cos(𝜃𝜃°)

𝜃𝜃, in degrees 210 225 240 270 300 315 330 360

cos(𝜃𝜃°)

𝜃𝜃, in degrees 390 405 420 450 480 495 510 540

cos(𝜃𝜃°)

B. Using the values in the table, sketch the graph of the sine function on the interval [0, 540].


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5. A. For the interval [0, 540], describe the values of 𝜃𝜃 at which the cosine function has relative maxima and minima.

B. For the interval l [0, 540], describe the values of 𝜃𝜃 for which the cosine function is increasing and decreasing.

C. For the interval [0, 540], list the values of 𝜃𝜃 at which the graph of the cosine function crosses the

horizontal axis.

D. Describe the end behavior of the cosine function.

E. Based on the graph, is cosine an odd function, even function, or neither? How do you know?

F. Describe how the cosine function repeats. 6. How are the sine function and the cosine function related to each other?


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Lesson Summary

• The sine and cosine curves are exactly the same shape (called a sinusoidal wave).

• Both y = sin x and y = cos x have a maximum of 1 and a minimum of -1.

• The sine curve goes through the origin, the cosine curve intersects the y-axis at 1.

• The curves continue all the way along the x-axis but the part between 0 and 360 is the basic shape which repeats itself again and again.

[source: http://www.bbc.co.uk/bitesize/standard/maths_ii/trigonometry/graphs/revision/1/]


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http://www.bbc.co.uk/bitesize/standard/maths_ii/trigonometry/graphs/revision/1/


Homework Problem Set 1. a. Graph the sine function on the interval [−360, 360] showing all key points of the graph (horizontal and

vertical intercepts and maximum and minimum points). Then, use the graph to answer each of the following questions.

b. On the interval [−360, 360], what are the relative minima of the sine function? Why?

c. On the interval [−360, 360], what are the relative maxima of the sine function? Why?

d. On the interval [−360, 360], for what values of 𝜃𝜃 is sin(𝜃𝜃°) = 0? Why?

e. If we continued to extend the graph in either direction, what would it look like? Why?

f. Arrange the following values in order from smallest to largest by using their location on the graph.

sin(170°) sin(85°) sin(−85°) sin(200°)

g. On the interval (90, 270), is the graph of the sine function increasing or decreasing? Based on that, name another interval not included in (90, 270) where the sine function must have the same behavior.

-1

-0.5

0

0.5

1

-360 -330 -300 -270 -240 -210 -180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180 210 240 270 300 330 360


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2. a. Graph the cosine function on the interval [−360, 360] showing all key points of the graph (horizontal and vertical intercepts and maximum and minimum points). Then, use the graph to answer each of the following questions.

b. On the interval [−360, 360], what are the relative minima of the cosine function? Why?

c. On the interval [−360, 360], what are the relative maxima of the cosine function? Why?

d. On the interval [−360, 360], for what values of 𝜃𝜃 is cos(𝜃𝜃°) = 0? Why?

e. If we continued to extend the graph in either direction, what would it look like? Why?

f. What can be said about the end behavior of the cosine function?

g. Arrange the following values in order from smallest to largest by using their location on the graph.

cos(135°) cos(85°) cos(−15°) cos(190°)

-1

-0.5

0

0.5

1

-360 -330 -300 -270 -240 -210 -180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180 210 240 270 300 330 360


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Hart Interactive – Algebra 2/Trigonometry

Unit 7 Vocabulary

Knowledge Rating*

Lesson Vocabulary


11 Radians The radian is a unit of __________ for angles. Whereas a full circle is 360 degrees, a full circle is _____ radians.

11 Central Angle A central angle is the angle that forms when two _____ meet at the __________ of a circle.

12 Identity

An identity is an equality relation A = B, such that A and B contain some variables and A and B produce the _____ value as each other regardless of what ________ are substituted for the variables.

12 Horizontal Shift or

Translation

A horizontal shift is a translation in which the _____ and _____ of a graph of a function is not changed, but the __________ of the graph is moved to the left or the right.

13 Amplitude

The amplitude is half the _______________ between the minimum and maximum values of the __________.

13 Period The period is the __________ required for the function to complete one full __________.


Module 3: Trigonometric Functions Unit 7: Transformations & Identities

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http://www.mathwords.com/m/minimum_of_a_function.htm

http://www.mathwords.com/m/maximum_of_a_function.htm

ALGEBRA 2/TRIG


Knowledge Rating*

Lesson Vocabulary


13 Frequency

The frequency is the __________ of the period. It tells you the number of __________ completed for each unit traveled from left to right.

13 & 14

Phase (Horizontal)

Shift (Translation)

The phase shift is the horizontal translation of a __________ function.

13 & 14

Vertical Translation

(Shift)

A vertical shift is a translation in which the size and shape of a graph of a function is not changed, but the location of the graph is moved to _____ or __________.

13 & 14 Midline

The midline is a __________ axis that is used as the _______________ line about which the graph of a periodic function oscillates.

16 Sinusoidal

A sinusoidal wave is a periodic function which __________ all its values after a certain period. A sinusoidal function or sinusoid function is generally the sine wave function.

17 Vertical Asymptote

Vertical asymptotes are straight lines of the form x = number, toward which a function f(x) __________ infinitesimally __________ , but never reaches the line, as f(x) increases without bound.



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ALGEBRA 2/TRIG


Knowledge Rating*

Lesson Vocabulary


18 Pythagorean Identities

sin2θ + cos2θ = _____

1 + cot2θ = _____θ

tan2θ + 1 = _____θ

20 Honors

Sum & Difference Identities

sin(a + b) = _________ + cos(a)sin(b)

cos(a + b) = cos(a)cos(b) – ________

sin(a – b) = __________– cos(a)sin(b)

cos(a – b) = cos(a)cos(b) + _________

20 Honors

Double Angle Identities

sin 2A = __________

cos 2A = __________

2

2tantan2

1 tanA

AA

=−

21 Honors

Parametric Functions

Rather than using a single equation to define two variables with respect to one another, parametric equations exist as a set that relates the two variables to one another with respect to a third variable, such as time.



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ALGEBRA 2/TRIG



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ALGEBRA 2/TRIG


Name: ____________________________________________________________________ Period: ______

Concept Circles

Directions: Of all the terms in Unit 7’s vocabulary list, choose five that are related to each other in some way. Write how they are related on the “Concept” line and write the five terms in the blank spaces of the circle. Then in each sector of the circle, write one statement that connects the term to the concept. Be prepared to share your Concept Circle with the class.


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ALGEBRA 2/TRIG



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Lesson 11: Awkward! Who Chose the Number 360, Anyway?

Opening Exercise Who came up with our current system of using 360° in a turn? The ancient Babylonians! They made many astronomical observations that led to the discovery of trigonometry. They are also the ones responsible for our system of measuring rotations and angles. It appears that the Babylonians subdivided the circle using the angle of an equilateral triangle as the basic unit. Since they used a base-60 number system, they divided each angle of the equilateral triangle into 60 smaller units, each with measure 1 degree, giving 360 degrees in a turn. Each degree is subdivided into 60 minutes, and each minute is subdivided into 60 seconds. For our purposes now, using 360° in a turn is cumbersome. Instead of basing our measurement system on an arbitrary number like 360, we will instead use a system in which the measures of angles and rotations are determined by the length of the corresponding arc of a unit circle.

1. Watch the YouTube video “What are Radians?”

(https://www.youtube.com/watch?v=cgPYLJ-s5II) and complete the questions below. A. Explain what the figure at the right is illustrating.

B. How many radians are in one complete rotation? C. How are radians related to degrees? D. How many degrees are in one radian?

Lesson 11: Awkward! Who Chose the Number 360, Anyway? Unit 7: Transformations & Identities

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https://www.youtube.com/watch?v=cgPYLJ-s5II


A circle is defined by a point and a radius. If we start with a circle of any radius and look at a sector of that circle with an arc length equal to the length of the radius, then the central angle of that sector is always the same size. We define a radian to be the measure of that central angle and denote it by 1 rad.

Thus, a radian measures how far one radius will wrap around the circle. For any circle, it takes 2𝜋𝜋 ≈ 6.3 radius lengths to wrap around the circumference. In the figure, 6 radius lengths are shown around the circle, with roughly 0.3 radius lengths left over.

2. Use a protractor that measures angles in degrees to find an approximate degree measure for an angle with measure 1 rad. Use one of the figures from the previous discussion. Does this agree with the video?

Converting From Degrees to Radians 3. Convert from degrees to radians: 45° 4. Convert from degrees to radians: 33°


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Converting From Radians to Degrees

5. Convert from radians to degrees: −π3

rad

6. Convert from radians to degrees: 19π17

rad

7. Complete the table below, converting from degrees to radians or from radians to degrees as necessary. Leave your answers in exact form, involving 𝜋𝜋.

Degrees Radians

45° 𝜋𝜋4

120°

−5𝜋𝜋6

3𝜋𝜋2

450°

𝑥𝑥°

𝑥𝑥


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Changing the Graph of sin(x)

8. If degrees are changed to radians, then one period of the graph of 𝑦𝑦 = sin(𝑥𝑥) on a grid with the same scale on the horizontal and vertical axes now looks like the graph below. Fill in the missing radian measures.

From this point forward, trigonometric functions will always be graphed using radians for measuring rotation instead of degrees. It turns out that radians make many calculations much easier in later work in mathematics. If there is no degree symbol or specification, then the use of radians is implied.

At the right is a unit circle with the radian measures given instead of the degree measures.


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Lesson Summary A radian is the measure of the central angle of a sector of a circle with arc length of one radius

length.

If there is no degree symbol or specification, then the use of radians is implied. There are 2𝜋𝜋 radians in a 360° rotation, also known as a turn, so degrees are converted

to radians and radians to degrees by: 2𝜋𝜋 rad = 1 turn = 360°.

From this point forward, all work will be done with radian measures exclusively for rotation and as the independent variables in the trigonometric functions. The diagram is nearly the same as the one for the sine and cosine functions in Lesson 4, but this time it is labeled with rotations measured in radians.

Example: Use the diagram to find cos�7𝜋𝜋6�

and sin �− 𝜋𝜋6�

cos �7𝜋𝜋6� = −√3

2

sin �− 𝜋𝜋6� = −1

2


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Homework Problem Set 1. Use a radian protractor to measure the amount of rotation in radians of ray 𝐵𝐵𝐵𝐵 to ray 𝐵𝐵𝐵𝐵 in the indicated

direction. Measure to the nearest 0.1 radian. Use negative measures to indicate clockwise rotation.

a.

b.

c.

d.

e.

f.

g.

h.


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2. Complete the table below, converting from degrees to radians. Where appropriate, give your answers in the form of a fraction of 𝜋𝜋.

Degrees Radians

90°

300°

−45°

−315°

−690°

334

°

90𝜋𝜋°

−45°𝜋𝜋

3. Complete the table below, converting from radians to degrees.

Radians Degrees 𝜋𝜋4

𝜋𝜋6

5𝜋𝜋12

11𝜋𝜋36

−7𝜋𝜋24

−11𝜋𝜋12

49𝜋𝜋

49𝜋𝜋3


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4. Use the unit circle diagram from the Lesson Summary and your knowledge of the six trigonometric functions to complete the table below. Give your answers in exact form, as either rational numbers or radical expressions.

𝜃𝜃 cos(𝜃𝜃) sin(𝜃𝜃) tan(𝜃𝜃) cot(𝜃𝜃) sec(𝜃𝜃) csc(𝜃𝜃)

𝜋𝜋3

3𝜋𝜋4

5𝜋𝜋6

0

−3𝜋𝜋4

−7𝜋𝜋6

−11𝜋𝜋

3


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5. Use the unit circle diagram from the end of the lesson and your knowledge of the sine, cosine, and tangent functions to complete the table below. Select values of 𝜃𝜃 so that 0 ≤ 𝜃𝜃 < 2𝜋𝜋.

𝜃𝜃 cos(𝜃𝜃) sin(𝜃𝜃) tan(𝜃𝜃)

12

−√3

−√22

1

−√22

√22

−1 0

0 −1

−12

√33

6. How many radians does the minute hand of a clock rotate through over 10 minutes? How many degrees?

7. How many radians does the minute hand of a clock rotate through over half an hour? How many degrees?

8. What is the radian measure of an angle subtended by an arc of a circle with radius 4 cm if the intercepted arc has length 14 cm? How many degrees?


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9. What is the radian measure of an angle formed by the minute and hour hands of a clock when the clock

reads 1:30? How many degrees? (Hint: You must take into account that the hour hand is not directly on the 1.)

10. What is the radian measure of an angle formed by the minute and hour hands of a clock when the clock reads 5:45? How many degrees?

11. How many degrees does the earth revolve on its axis each hour? How many radians? 12. The distance from the equator to the North Pole is almost exactly 10,000 km.

a. Roughly how many kilometers is 1 degree of latitude?

b. Roughly how many kilometers is 1 radian of latitude?


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Lesson 12: Basic Trigonometric Identities from Graphs

Exploratory Challenge 1

You will need: colored pencils

Consider the function 𝑓𝑓(𝑥𝑥) = sin(𝑥𝑥) where 𝑥𝑥 is measured in radians.

1. A. Graph 𝑓𝑓(𝑥𝑥) = sin(𝑥𝑥) on the interval [−𝜋𝜋, 5𝜋𝜋] by constructing a table of values. Include all intercepts, relative maximum points, and relative minimum points of the graph. Then, use the graph to answer the questions that follow.

𝑥𝑥

𝑓𝑓(𝑥𝑥)

B. Using one of your colored pencils, mark the point on the graph at each of the following pairs of 𝑥𝑥-values.

𝑥𝑥 = −𝜋𝜋2

and 𝑥𝑥 = −𝜋𝜋2

+ 2𝜋𝜋

𝑥𝑥 = 𝜋𝜋 and 𝑥𝑥 = 𝜋𝜋 + 2𝜋𝜋

𝑥𝑥 = 7𝜋𝜋4

and 𝑥𝑥 = 7𝜋𝜋4

+ 2𝜋𝜋

C. What can be said about the 𝑦𝑦-values for each pair of 𝑥𝑥-values marked on the graph?

Lesson 12: Basic Trigonometric Identities from Graphs Unit 7: Transformations & Identities

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D. Will this relationship hold for any two 𝑥𝑥-values that differ by 2𝜋𝜋? Explain how you know.

E. Based on these results, make a conjecture by filling in the blank below.

For any real number 𝑥𝑥, sin(𝑥𝑥 + 2𝜋𝜋) = ______________________.

F. Test your conjecture by selecting another 𝑥𝑥-value from the graph and demonstrating that the equation from Part E holds for that value of 𝑥𝑥.

G. How does the conjecture in Part E support the claim that the sine function is a periodic function?

H. Use this identity to evaluate sin �13𝜋𝜋6�.

I. Using one of your colored pencils, mark the point on the graph at each of the following pairs of 𝑥𝑥-values.



+ 𝜋𝜋

𝑥𝑥 = 2𝜋𝜋 and 𝑥𝑥 = 2𝜋𝜋 + 𝜋𝜋



+ 𝜋𝜋

J. What can be said about the 𝑦𝑦-values for each pair of 𝑥𝑥-values marked on the graph?

K. Will this relationship hold for any two 𝑥𝑥-values that differ by 𝜋𝜋? Explain how you know.


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L. Based on these results, make a conjecture by filling in the blank below.

For any real number 𝑥𝑥, sin(𝑥𝑥 + 𝜋𝜋) = ______________________.

M. Test your conjecture by selecting another 𝑥𝑥-value from the graph and demonstrating that the equation from Part L holds for that value of 𝑥𝑥.

N. Is the following statement true or false? Use the conjecture from Part L to explain your answer.

sin �4π3� = −sin �

𝜋𝜋3�

O. Using one of your colored pencils, mark the point on the graph at each of the following pairs of 𝑥𝑥-values.

𝑥𝑥 = −3𝜋𝜋4



and 𝑥𝑥 = 𝜋𝜋2

P. What can be said about the 𝑦𝑦-values for each pair of 𝑥𝑥-values marked on the graph?

Q. Will this relationship hold for any two 𝑥𝑥-values with the same magnitude but opposite sign? Explain how you know.

R. Based on these results, make a conjecture by filling in the blank below.

For any real number 𝑥𝑥, sin(−𝑥𝑥) = ______________________. S. Test your conjecture by selecting another 𝑥𝑥-value from the graph and demonstrating that the

equation from Part R holds for that value of 𝑥𝑥.


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T. Is the sine function an odd function, even function, or neither? Use the identity from Part R to explain.

U. Describe the 𝑥𝑥-intercepts of the graph of the sine function.

V. Describe the end behavior of the sine function.


Consider the function 𝑔𝑔(𝑥𝑥) = cos(𝑥𝑥) where 𝑥𝑥 is measured in radians.

2. A. Graph 𝑔𝑔(𝑥𝑥) = cos(𝑥𝑥) on the interval [−𝜋𝜋, 5𝜋𝜋] by constructing a table of values. Include all intercepts, relative maximum points, and relative minimum points. Then, use the graph to answer the questions that follow.

𝑥𝑥

𝑔𝑔(𝑥𝑥)


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B. Using one of your colored pencils, mark the point on the graph at each of the following pairs of 𝒙𝒙-values.



+ 2𝜋𝜋

𝑥𝑥 = 𝜋𝜋 and 𝑥𝑥 = 𝜋𝜋 + 2𝜋𝜋



+ 2𝜋𝜋

C. What can be said about the 𝑦𝑦-values for each pair of 𝑥𝑥-values marked on the graph?

D. Will this relationship hold for any two 𝑥𝑥-values that differ by 2𝜋𝜋? Explain how you know.

E. Based on these results, make a conjecture by filling in the blank below.

For any real number 𝑥𝑥, cos(𝑥𝑥 + 2𝜋𝜋) = _______________.

F. Test your conjecture by selecting another 𝑥𝑥-value from the graph and demonstrating that the equation from Part E holds for that value of 𝑥𝑥.

G. How does the conjecture from Part E support the claim that the cosine function is a periodic function?

H. Use this identity to evaluate cos �9𝜋𝜋4�.


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I. Using one of your colored pencils, mark the point on the graph at each of the following pairs of 𝑥𝑥-values.



+ 𝜋𝜋

𝑥𝑥 = 2𝜋𝜋 and 𝑥𝑥 = 2𝜋𝜋 + 𝜋𝜋



+ 𝜋𝜋

J. What can be said about the 𝑦𝑦-values for each pair of 𝑥𝑥-values marked on the graph?

K. Will this relationship hold for any two 𝑥𝑥-values that differ by 𝜋𝜋? Explain how you know.

L. Based on these results, make a conjecture by filling in the blank below.

For any real number 𝑥𝑥, cos (𝑥𝑥 + 𝜋𝜋) = _______________.

M. Test your conjecture by selecting another 𝑥𝑥-value from the graph and demonstrating that the equation from Part L holds for that value of 𝑥𝑥.

N. Is the following statement true or false? Use the identity from Part L to explain your answer.

cos�5𝜋𝜋3� = −cos �

2𝜋𝜋3�

O. Using one of your colored pencils, mark the point on the graph at each of the following pairs of 𝑥𝑥-values.

𝑥𝑥 = −3𝜋𝜋4


𝑥𝑥 = −𝜋𝜋 and 𝑥𝑥 = 𝜋𝜋


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P. What can be said about the 𝑦𝑦-values for each pair of 𝑥𝑥-values marked on the graph?

Q. Will this relationship hold for any two 𝑥𝑥-values with the same magnitude and same sign? Explain how you know.

R. Based on these results, make a conjecture by filling in the blank below.

For any real number , cos(−𝑥𝑥) = _______________.

S. Test your conjecture by selecting another 𝑥𝑥-value from the graph and demonstrating that the identity is true for that value of 𝑥𝑥.

T. Is the cosine function an odd function, even function, or neither? Use the identity from Part O to explain.

U. Describe the 𝑥𝑥-intercepts of the graph of the cosine function.

V. Describe the end behavior of 𝑔𝑔(𝑥𝑥) = cos(𝑥𝑥).


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3. A. Graph both 𝑓𝑓(𝑥𝑥) = sin(𝑥𝑥) and 𝑔𝑔(𝑥𝑥) = cos(𝑥𝑥) on the graph below. Then, use the graphs to answer the questions that follow.

B. List ways in which the graphs of the sine and cosine functions are alike.

C. List ways in which the graphs of the sine and cosine functions are different.

D. What type of transformation would be required to make the graph of the sine function coincide with the graph of the cosine function?


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E. What is the smallest possible horizontal translation required to make the graph of 𝑓𝑓(𝑥𝑥) = sin(𝑥𝑥) coincide with the graph of 𝑔𝑔(𝑥𝑥) = cos(𝑥𝑥)?

F. What is the smallest possible horizontal translation required to make the graph of 𝑔𝑔(𝑥𝑥) = cos(𝑥𝑥) coincide with the graph of 𝑓𝑓(𝑥𝑥) = sin(𝑥𝑥)?

G. Use your answers from Parts E and F to fill in the blank below.

For any real number 𝑥𝑥, ________________ = cos �𝑥𝑥 − 𝜋𝜋2�.

For any real number 𝑥𝑥, ________________ = sin �𝑥𝑥 + 𝜋𝜋2�.


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Lesson Summary

TRIGONOMETRIC IDENTITIES

For all real numbers 𝑥𝑥:

Period Identities

sin(𝑥𝑥 + 2𝜋𝜋) = sin(𝑥𝑥) cos(𝑥𝑥 + 2𝜋𝜋) = cos(𝑥𝑥)

Horizontal Shift Identities

sin(𝑥𝑥 + 𝜋𝜋) = − sin(𝑥𝑥) cos(𝑥𝑥 + 𝜋𝜋) = − cos(𝑥𝑥)

sin �𝑥𝑥 + 𝜋𝜋2� = cos (𝑥𝑥) cos �𝑥𝑥 − 𝜋𝜋

2� = sin(𝑥𝑥)

Even/Odd Identities

sin(−𝑥𝑥) = − sin(𝑥𝑥) cos(−𝑥𝑥) = cos(𝑥𝑥)

Reciprocal Identities

1sin( )

csc( )x

x= 1

csc( )sin( )

xx

=

1cos( )

sec( )x

x= 1

sec( )cos( )

xx

=

1tan( )

cot( )x

x= 1

cot( )tan( )

xx

=


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Homework Problem Set 1. Describe the values of 𝑥𝑥 for which each of the following is true.

a. The cosine function has a relative maximum.

b. The sine function has a relative maximum.

2. Without using a calculator, rewrite each of the following in order from least to greatest. Use the graph to explain your reasoning.

sin �𝜋𝜋4� sin �− 2𝜋𝜋

3� sin �𝜋𝜋

4� sin �− 𝜋𝜋

2�


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3. Without using a calculator, rewrite each of the following in order from least to greatest. Use the graph to explain your reasoning.

cos �𝜋𝜋2� cos �5𝜋𝜋

4� cos �𝜋𝜋

4� cos(5𝜋𝜋)

4. Evaluate each of the following without a calculator using a trigonometric identity when needed.

cos �𝜋𝜋6� cos �−

𝜋𝜋6� cos�

7𝜋𝜋6� cos �

13𝜋𝜋6�

5. Evaluate each of the following without a calculator using a trigonometric identity when needed.

sin�3𝜋𝜋4� sin�

11𝜋𝜋4� sin�

7𝜋𝜋4� sin�

−5𝜋𝜋4

�


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6. Use the rotation through θ radians shown to answer each of the following questions. a. Explain why sin(−𝜃𝜃) = − sin(𝜃𝜃) for all real numbers 𝜃𝜃.

b. What symmetry does this identity demonstrate about the graph of 𝑦𝑦 = sin(𝑥𝑥)?

7. Use the same rotation shown in Problem 6 to answer each of the following questions. a. Explain why cos(−𝜃𝜃) = cos(𝜃𝜃).

b. What symmetry does this identity demonstrate about the graph of 𝑦𝑦 = cos(𝑥𝑥)?


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8. Find equations of two different functions that can be represented by the graph shown below—one sine and one cosine—using different horizontal transformations.

9. Find equations of two different functions that can be represented by the graph shown below—one sine and one cosine—using different horizontal translations.


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Hart Interactive – Algebra 2/Trig M3 Lesson 13

Lesson 13: Transforming the Graph of the Sine Function – A and ω

Opening Exercise

In Algebra 1 and Geometry you explored transformations of equations and shapes. In this lesson, you and your partner will look at the changes that can occur in trigonometric functions. We’ll focus on the sine function, but all of these transformation rules you’ll learn also apply to the other trig functions.

You will need a graphing utility.

1. Explore each parameter in the sinusoidal function 𝑓𝑓(𝑥𝑥) = 𝐴𝐴 sin(𝜔𝜔𝑥𝑥). Select several different values for the parameter, and explore the effects of changing the parameter’s value on the graph of the function compared to the graph of 𝑓𝑓(𝑥𝑥) = sin(𝑥𝑥). Record your observations in the table below. Include written descriptions and/or sketches of graphs.

𝑨𝑨 Changes

𝑓𝑓(𝑥𝑥) = 𝐴𝐴 sin(𝑥𝑥)

Suggested 𝐴𝐴 values:

2, 3, 10, 0,−1,−2, 12

, 15

,−13

𝝎𝝎 Changes

𝑓𝑓(𝑥𝑥) = sin(𝜔𝜔𝑥𝑥)

Suggested 𝜔𝜔 values:

2, 3, 5, 12

, 14

, 0,−1,−2,𝜋𝜋, 2𝜋𝜋, 3𝜋𝜋, 𝜋𝜋2

, 𝜋𝜋4

When A > 1: When ω > 1:

When 0 < A < 1: When 0 < ω < 1:

When A < 0: When ω < 0:

When A = 1: When ω = 1:

Changes to A change: Changes to ω change:

The maximum and minimum values of the graph are: The number of cycles of the graph:

When A = 0: When ω = 0:

Lesson 13: Transforming the Graph of the Sine Function – A and ω Unit 7: Transformations & Identities

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For each equation below, explain the change to the parent graph 𝒇𝒇(𝒙𝒙) = 𝐬𝐬𝐬𝐬𝐬𝐬(𝒙𝒙). Check your answer by graphing each equation on your graphing utility.

2. f(x) = 5sin(x) 3. f(x) = ―3sin(x) 4. f(x) = ― 12

sin(x) 5. f(x) = 23

sin(x)

6. f(x) = sin(5x) 7. f(x) = sin(3x) 8. f(x) = 1sin

2x

9. f(x) = 2sin

3x

Use the given grid for both questions below.

10. What will the result of an A value of 4 and an ω value of 2 on the same graph of sine? Explain, write the equation, and then sketch the graph. Use your graphing utility to check your graph.

11. What will the result of an A value of -2 and an ω value of ½ on the same graph of sine? Explain, write the equation, and then sketch the graph. Use your graphing utility to check your graph.


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Lesson Summary

This lesson investigated the effects of the parameters 𝐴𝐴 and 𝜔𝜔 on the graph of the function 𝑓𝑓(𝑥𝑥) = 𝐴𝐴 sin(𝜔𝜔𝑥𝑥).

The amplitude of the function is |𝐴𝐴|; the vertical distance from a maximum point to the midline of the graph is |𝐴𝐴|.

The frequency of the function is 𝑓𝑓 = |𝜔𝜔|2𝜋𝜋, and the period is 𝑃𝑃 = 2𝜋𝜋

|𝜔𝜔|. The period is the vertical

distance between two consecutive maximal points on the graph of the function.

These parameters affect the graph of 𝑓𝑓(𝑥𝑥) = 𝐴𝐴 cos(𝜔𝜔𝑥𝑥) similarly.

Homework Problem Set

State the period and amplitude for each equation.

1. f(x) = 7cos(3x) 2. f(x) = 6sin(2x) 3. f(x) = -4cos(x)

4. f(x) = -sin(2πx) 5. f(x) = 3sin(πx) 6. f(x) = -2cos(3πx)


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Matching – Match each equation with its graph. All graphs are scaled from [-2π, 2π] on the x-axis.

7. f(x) = 2 sin(x) 8. f (x) = -1 sin(x) 9. f (x) = -2sin(x)

10. f (x) = sin(x) 11. f (x) = sin(2x) 12. f (x) = sin( 12x)

13. f (x) = sin(3x) 14. f (x) = 2sin(3x) 15. f (x) = 3sin(2x)

A.

B.

C.

D.

E.

F.

G.

H.

I.


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Graph each equation.

16. f(x) = 7 sin(x) 17. f(x) = sin(2x)

18. f(x) = -4 sin(2x) 19. f(x) =- 4sin(12

x)


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Write the equation of each graph below.

20.

21.


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Lesson 14: Transform the Graph of the Sine Function with h and k

Opening Exercise

In the last lesson, your focus was on the changes on the amplitude, A, and on the period change caused by ω. In this lesson you’ll look at the changes that occur with vertical and horizontal translations.

You will need a graphing utility.

1. Explore your assigned parameter in the sinusoidal function 𝑓𝑓(𝑥𝑥) = sin(𝑥𝑥 − ℎ) + 𝑘𝑘. Select several different values for your assigned parameter, and explore the effects of changing the parameter’s value on the graph of the function compared to the graph of 𝑓𝑓(𝑥𝑥) = sin(𝑥𝑥). Record your observations in the table below. Include written descriptions and/or sketches of graphs.

𝒌𝒌 Changes

𝑓𝑓(𝑥𝑥) = sin(𝑥𝑥) + 𝑘𝑘 Suggested 𝑘𝑘 values: 2, 3, 10, 0,−1,−2, 1

2, 15

,−13

𝒉𝒉 Changes

𝑓𝑓(𝑥𝑥) = sin(𝑥𝑥 − ℎ) Suggested ℎ values: 𝜋𝜋,−𝜋𝜋, 𝜋𝜋

2,−𝜋𝜋

4, 2𝜋𝜋, 2, 0,−1,−2, 5,−5

When k > 0: When h > 0:

When k < 0: When h < 0:

When k = 0: When h = 0:

Changes to k change control: Changes to h change control:

Lesson 14: Transforming the Graph of the Sine Function – h and k Unit 7: Transformations & Identities

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For each equation below, explain the change to the parent graph 𝒇𝒇(𝒙𝒙) = 𝐬𝐬𝐬𝐬𝐬𝐬(𝒙𝒙). Check your answer by graphing each equation on your graphing utility.

2. f(x) = sin(x) + 5 3. f(x) = sin(x) – 2 4. f(x) = sin(x) + 7 5. f(x) = sin(x) – 1

6. f(x) = sin(x – 3π ) 7. f(x) = sin(x + π) 8. f(x) = sin(x – 2) + 3 9. f(x) = sin(x + 2) – 1

Use the given grid for both questions below.

10. What will the result of an h value of π and an k value of 2 on the same graph of sine? Explain, write the equation, and then sketch the graph. Use your graphing utility to check your graph.

11. What will the result of an h value of π/2 and an k value of -2 on the same graph of sine? Explain, write the equation, and then sketch the graph. Use your graphing utility to check your graph.


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Lesson Summary

This lesson investigated the effects of the parameters ℎ, and 𝑘𝑘 on the graph of the function

𝑓𝑓(𝑥𝑥) = sin(𝑥𝑥 − ℎ) + 𝑘𝑘.

The phase shift is ℎ. The value of ℎ determines the horizontal translation of the graph from the graph of the sine function. If ℎ > 0, the graph is translated ℎ units to the right, and if ℎ < 0, the graph is translated ℎ units to the left.

The graph of 𝑦𝑦 = 𝑘𝑘 is the midline. The value of 𝑘𝑘 determines the vertical translation of the graph compared to the graph of the sine function. If 𝑘𝑘 > 0, then the graph shifts 𝑘𝑘 units upward. If 𝑘𝑘 < 0, then the graph shifts 𝑘𝑘 units downward.

These parameters affect the graph of 𝑓𝑓(𝑥𝑥) = cos(𝑥𝑥 − ℎ) + 𝑘𝑘 similarly.

Homework Problem Set State the phase shift and vertical shift for each equation.

1. f(x) = cos(x + π/2) + 4 2. f(x) = sin(x – π) – 1 3. f(x) = cos(x) + 7

4. f(x) = sin(x + π) 5. f(x) = sin(x – π/4) 6. f(x) = cos(x + 3π/4) – 2


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Matching – Match each equation with its graph. All graphs are scaled from [-2π, 2π] on the x-axis.

7. f(x) = sin(x) – 2 8. f (x) = sin(x) + 1 9. f (x) = sin(x) – 1

10. f (x) = sin(x) 11. f (x) = sin(x – π/2) 12. f (x) = sin(x + π)

13. f (x) = sin(x – π/2) – 2 14. f (x) = sin(x + π) + 1 15. f (x) = sin(x + π/2) – 1

A.

B.

C.

D.

E.

F.

G.

H.

I.


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Graph each equation.

16. f(x) = sin(x + π) – 2 17. f(x) = sin(x – π/2) + 4

18. f(x) = sin(x) + 3 19. f(x) =sin(x – π)


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Write the equation of each graph below.

20.

21.


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Lesson 15: Transforming the Graph of the Sine Function – Putting It All Together

Opening Exercise

Now that you’ve looked at the four ways to transform a trigonometric function, it is time to put it all together. We’ll start with a Desmos activity, Trigonometric Graphing.

1. Activity – Desmos Trigonometric Graphing

. . .


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Let’s look at how we put all these pieces together to create one graph of a complex trigonometric function.

2. Before we graph the following function: 𝑓𝑓(𝑥𝑥) = 3 sin�4 �𝑥𝑥 − 𝜋𝜋6�� + 2. Let’s look at all the parts of the equation.

A. What are the values of the parameters 𝜔𝜔, ℎ, 𝐴𝐴, and 𝑘𝑘, and what do they mean?

B. First, we consider the parameter 𝜔𝜔, which affects both period and frequency. What happens when 𝜔𝜔 = 4? Graph 𝑓𝑓(𝑥𝑥) = sin(4𝑥𝑥) on the grid at the right.

C. Next, we examine the horizontal translation by ℎ = 𝜋𝜋

6. What effect does that have on the graph?

Graph 𝑓𝑓(𝑥𝑥) = sin�4 �𝑥𝑥 − 𝜋𝜋6�� on the grid at the

right. Notice that we are still using ω = 4.


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D. The next step is to look at the effect of 𝐴𝐴 = 3. What does that do to our graph? How does it compare to the previous graph? Graph

𝑓𝑓(𝑥𝑥) = 3sin�4 �𝑥𝑥 − 𝜋𝜋6�� on the grid on the

right.

E. The final change is to consider the effect of 𝑘𝑘 = 2. This gives us the final graph, which is the graph of the function 𝑓𝑓(𝑥𝑥) =3 sin �4 �𝑥𝑥 − 𝜋𝜋

6� � + 2. What happens to the

graph of this function when 𝑘𝑘 = 2?


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Practice Exercises 3 – 10

For each function, indicate the amplitude, frequency, period, phase shift, vertical translation, and equation of the midline. Graph the function together with a graph of the sine function 𝑓𝑓(𝑥𝑥) = sin(𝑥𝑥) on the same axes. Graph at least one full period of each function.

3. 𝑔𝑔(𝑥𝑥) = 3 sin(2𝑥𝑥) − 1

4. 𝑔𝑔(𝑥𝑥) = 12 sin �1

4 (𝑥𝑥 + 𝜋𝜋)�


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5. 𝑔𝑔(𝑥𝑥) = 5 sin(−2𝑥𝑥) + 2

6. 𝑔𝑔(𝑥𝑥) = −2 sin �3 �𝑥𝑥 + 𝜋𝜋6��


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7. 𝑔𝑔(𝑥𝑥) = 3 sin(𝑥𝑥 + 𝜋𝜋) + 3

8. 𝑔𝑔(𝑥𝑥) = − 23 sin(4𝑥𝑥) − 3


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9. 𝑔𝑔(𝑥𝑥) = 𝜋𝜋 sin �𝑥𝑥2� + 𝜋𝜋

10. 𝑔𝑔(𝑥𝑥) = 4 sin �12 (𝑥𝑥 − 5𝜋𝜋)�


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Homework Problem Set For each function, indicate the amplitude, frequency, period, phase shift, and vertical translations, and equation of the midline.

Equation Amplitude Frequency Period Phase Shift

Vertical Translations

Equation of the

Midline

1. 𝑔𝑔(𝑥𝑥) = 3 sin �𝑥𝑥 − 𝜋𝜋4�

2. 𝑔𝑔(𝑥𝑥) = 5 sin(4𝑥𝑥)

3. 𝑔𝑔(𝑥𝑥) = 4 sin �3 �𝑥𝑥 + 𝜋𝜋2��

4. 𝑔𝑔(𝑥𝑥) = 6 sin(2𝑥𝑥 + 3𝜋𝜋) (Hint: First, rewrite the function in the form

𝑔𝑔(𝑥𝑥) = 𝐴𝐴 sin�𝜔𝜔(𝑥𝑥 − ℎ)�. )

5. 𝑔𝑔(𝑥𝑥) = cos(3𝑥𝑥)

6. 𝑔𝑔(𝑥𝑥) = cos �𝑥𝑥 − 3𝜋𝜋4 �

7. 𝑔𝑔(𝑥𝑥) = 3 cos �𝑥𝑥4�

8. 𝑔𝑔(𝑥𝑥) = 3 cos(2𝑥𝑥) − 4

9. 𝑔𝑔(𝑥𝑥) = 4 cos �𝜋𝜋4 − 2𝑥𝑥�

(Hint: First, rewrite the function in the form 𝑔𝑔(𝑥𝑥) = 𝐴𝐴 cos�𝜔𝜔(𝑥𝑥 − ℎ)�.)


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For each problem, sketch the graph of the pairs of indicated functions on the same set of axes without using a calculator or other graphing technology.

10. 𝑓𝑓(𝑥𝑥) = sin(4𝑥𝑥), 𝑔𝑔(𝑥𝑥) = sin(4𝑥𝑥) + 2 11. 𝑓𝑓(𝑥𝑥) = sin �12 𝑥𝑥�, 𝑔𝑔(𝑥𝑥) = 3 sin �1

2𝑥𝑥�

12. 𝑓𝑓(𝑥𝑥) = sin(−2𝑥𝑥), 𝑔𝑔(𝑥𝑥) = sin(−2𝑥𝑥) − 3 13. 𝑓𝑓(𝑥𝑥) = 3 sin(𝑥𝑥), 𝑔𝑔(𝑥𝑥) = 3 sin �𝑥𝑥 − 𝜋𝜋2�


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14. 𝑓𝑓(𝑥𝑥) = −4 sin(𝑥𝑥), 𝑔𝑔(𝑥𝑥) = −4 sin �13 𝑥𝑥� 15. 𝑓𝑓(𝑥𝑥) = 3

4 sin(𝑥𝑥), 𝑔𝑔(𝑥𝑥) = 34 sin(𝑥𝑥 − 1)

16. 𝑓𝑓(𝑥𝑥) = sin(2𝑥𝑥), 𝑔𝑔(𝑥𝑥) = sin �2 �𝑥𝑥 − 𝜋𝜋6�� 17. 𝑓𝑓(𝑥𝑥) = 4sin(𝑥𝑥) − 3, 𝑔𝑔(𝑥𝑥) = 4 sin �𝑥𝑥 − 𝜋𝜋

4� − 3


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CHALLENGE PROBLEMS

18. Show that if the graphs of the functions 𝑓𝑓(𝑥𝑥) = 𝐴𝐴 sin�𝜔𝜔(𝑥𝑥 − ℎ1)� + 𝑘𝑘 and 𝑔𝑔(𝑥𝑥) = 𝐴𝐴 sin�𝜔𝜔(𝑥𝑥 − ℎ2)� + 𝑘𝑘 are the same, then ℎ1 and ℎ2 differ by an integer multiple of the period.

19. Show that if ℎ1 and ℎ2 differ by an integer multiple of the period, then the graphs of 𝑓𝑓(𝑥𝑥) = 𝐴𝐴 sin�𝜔𝜔(𝑥𝑥 − ℎ1)� + 𝑘𝑘 and 𝑔𝑔(𝑥𝑥) = 𝐴𝐴 sin�𝜔𝜔(𝑥𝑥 − ℎ2)� + 𝑘𝑘 are the same graph.

20. Find the 𝑥𝑥-intercepts of the graph of the function 𝑓𝑓(𝑥𝑥) = 𝐴𝐴 sin�𝜔𝜔(𝑥𝑥 − ℎ)� in terms of the period 𝑃𝑃, where 𝜔𝜔 > 0.


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Lesson 16: Tides, Sound Waves, and Stock Markets – Modeling with Sinusoidal Graphs

Opening Exercise

Over the past few years, you’ve used linear, quadratic, exponential and logarithmic models to represent real world data. Sine and cosine graphs are also good tools for modeling data. Ocean tides, sound wave and the stock market are just a few of the applications where a sinusoidal equation is the best fit. Sometimes you won’t know which model to use until you graph the data.

1. You are working on a team analyzing the following data gathered by your colleagues:

(−1.1, 5), (0, 105), (1.5, 178), (4.3, 120)

A. Your coworker Alexandra says that the model you should use to fit the data is

𝑘𝑘(𝑡𝑡) = 100 ∙ sin(1.5𝑡𝑡) + 105.

Sketch Alexandra’s model on the axes below.

B. Another teammate Randall says that the model you should use to fit the data is 𝑔𝑔(𝑡𝑡) = −16𝑡𝑡2 + 72𝑡𝑡 + 105.

Sketch Randall’s model on the axes below

How does the graph of Alexandra’s model 𝑘𝑘(𝑡𝑡) = 100 ∙ sin(1.5𝑡𝑡) + 105 relate to the four points? Is her model a good fit to this data?

How does the graph of Randall’s model 𝑔𝑔(𝑡𝑡) = −16𝑡𝑡2 + 72𝑡𝑡 + 105 relate to the four points? Is his model a good fit to the data?

Lesson 16: Tides, Sound Waves, and Stock Markets – Modeling with Sinusoidal Graphs Unit 7: Transformations & Identities

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Discussion 2. Suppose the four points represent positions of a projectile fired into the air. Which of the two models is

more appropriate in that situation, and why?

3. In general, how do we know which model to choose?

4. What are the characteristics of a nonconstant linear function? 5. What are the characteristics of a quadratic function?

6. What are the characteristics of a sinusoidal function?

7. What are the characteristics of an exponential function? 8. What are the clues in the context of a particular situation that suggest the use of a particular type of

function as a model?


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Practice Exercises

9. A. The table below contains the number of daylight hours in Oslo, Norway, on the specified dates. Plot the data on the grid provided and decide how to best represent it.

B. Looking at the data, what type of function appears to be the best fit?

C. Looking at the context in which the data was gathered, what type of function should be used to model the data?

D. Do you have enough information to find a model that is appropriate for this data? Either find a model or explain what other information you would need to do so.

Date Hours and Minutes Hours August 1 16: 56 16.93

September 1 14: 15 14.25 October 1 11: 33 11.55

November 1 8: 50 8.83


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10. The goal of the U.S. Centers for Disease Control and Prevention (CDC) is to protect public health and safety through the control and prevention of disease, injury, and disability. Suppose that 45 people have been diagnosed with a new strain of the flu virus and that scientists estimate that each person with the virus will infect 5 people every day with the flu. A. What type of function should the scientists at the CDC use to model the initial spread of this strain of

flu to try to prevent an epidemic? Explain how you know.

B. Do you have enough information to find a model that is appropriate for this situation? Either find a model or explain what other information you would need to do so.

11. An artist is designing posters for a new advertising campaign. The first poster takes 10 hours to design, but each subsequent poster takes roughly 15 minutes less time than the previous one as he gets more practice. A. What type of function models the amount of time needed to create 𝑛𝑛 posters, for 𝑛𝑛 ≤ 20? Explain

how you know.



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12. A homeowner notices that her heating bill is the lowest in the month of August and increases until it reaches its highest amount in the month of February. After February, the amount of the heating bill slowly drops back to the level it was in August, when it begins to increase again. The amount of the bill in February is roughly four times the amount of the bill in August. A. What type of function models the amount of the heating bill in a particular month? Explain how you

know.


13. An online merchant sells used books for $5.00 each, and the sales tax rate is 6% of the cost of the books. Shipping charges are a flat rate of $4.00 plus an additional $1.00 per book. A. What type of function models the total cost, including the shipping costs, of a purchase of 𝑥𝑥 books?

Explain how you know. B. Do you have enough information to find a model that is appropriate for this situation? Either find a

model or explain what other information you would need to do so.


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14. A stunt woman falls from a tall building in an action-packed movie scene. Her speed increases by 32 ft/s for every second that she is falling. A. What type of function models her distance from the ground at time 𝑡𝑡 seconds? Explain how you

know.


Lesson Summary

If we expect from the context that each new term in the sequence of data is a constant added to the previous term, then we try a linear model.

If we expect from the context that the second differences of the sequence are constant (meaning that the rate of change between terms either grows or shrinks linearly), then we try a quadratic model.

If we expect from the context that each new term in the sequence of data is a constant multiple of the previous term, then we try an exponential model.

If we expect from the context that the sequence of terms is periodic, then we try a sinusoidal model.

Model Equation of Function Rate of Change Linear 𝑓𝑓(𝑡𝑡) = 𝑎𝑎𝑡𝑡 + 𝑏𝑏 for 𝑎𝑎 ≠ 0 Constant

Quadratic 𝑔𝑔(𝑡𝑡) = 𝑎𝑎𝑡𝑡2 + 𝑏𝑏𝑡𝑡 + 𝑐𝑐 for 𝑎𝑎 ≠0 Changing linearly

Exponential ℎ(𝑡𝑡) = 𝑎𝑎𝑏𝑏𝑐𝑐𝑐𝑐 for 0 < 𝑏𝑏 < 1

or 𝑏𝑏 > 1 A multiple of the

current value

Sinusoidal 𝑘𝑘(𝑡𝑡) = 𝐴𝐴 sin�𝑤𝑤(𝑡𝑡 − ℎ)� + 𝑘𝑘

for 𝐴𝐴,𝑤𝑤 ≠ 0 Periodic


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Homework Problem Set 1. Find equations of both a sine function and a cosine function that could each represent the graph given below.

2. Find equations of both a sine function and a cosine function that could each represent the graph given below.


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3. Rapidly vibrating objects send pressure waves through the air that are detected by our ears and then interpreted by our brains as sound. Our brains analyze the amplitude and frequency of these pressure waves.

A speaker usually consists of a paper cone attached to an electromagnet. By sending an oscillating electric current through the electromagnet, the paper cone can be made to vibrate. By adjusting the current, the amplitude and frequency of vibrations can be controlled.

The following graph shows the pressure intensity (𝐼𝐼) as a function of time (𝑥𝑥), in seconds, of the pressure waves emitted by a speaker set to produce a single pure tone.

a. Does it seem more natural to use a sine or a cosine function to fit to this graph? Explain your thinking.

b. Find the equation of a trigonometric function that fits this graph.

4. A new car depreciates at a rate of about 20% per year, meaning that its resale value decreases by roughly 20%

each year. After hearing this, Brett said that if you buy a new car this year, then after 5 years the car has a resale value of $0.00. Is his reasoning correct? Explain how you know.


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5. Alexei just moved to Seattle, and he keeps track of the average rainfall for a few months to see if the city deserves its reputation as the rainiest city in the United States.

Month Average rainfall July 0.93 in.

September 1.61 in. October 3.24 in.

December 6.06 in.

What type of function should Alexei use to model the average rainfall in month 𝑡𝑡? 6. Sunny, who wears her hair long and straight, cuts her hair once per year on January 1, always to the same

length. Her hair grows at a constant rate of 2 cm per month. Is it appropriate to model the length of her hair with a sinusoidal function? Explain how you know.

7. On average, it takes 2 minutes for a customer to order and pay for a cup of coffee.

a. What type of function models the amount of time you wait in line as a function of how many people are in front of you? Explain how you know.

b. Find a model that is appropriate for this situation.


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8. An online ticket-selling service charges $50.00 for each ticket to an upcoming concert. In addition, the buyer must pay 8% sales tax and a convenience fee of $6.00 for the purchase.

a. What type of function models the total cost of the purchase of 𝑛𝑛 tickets in a single transaction?

b. Find a model that is appropriate for this situation. 9. In a video game, the player must earn enough points to pass one level and progress to the next as shown

in the table below.

To pass this level… You need this many total points… 1 5,000 2 15,000 3 35,000 4 65,000

That is, the increase in the required number of points increases by 10,000 points at each level. a. What type of function models the total number of points you need to pass to level 𝑛𝑛? Explain how

you know.

b. Find a model that is appropriate for this situation.


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10. The southern white rhinoceros reproduces roughly once every three years, giving birth to one calf each time. Suppose that a nature preserve houses 100 white rhinoceroses, 50 of which are female. Assume that half of the calves born are female and that females can reproduce as soon as they are 1 year old.

a. What type of function should be used to model the population of female white rhinoceroses in the preserve?

b. Assuming that there is no death in the rhinoceros population, find a function to model the population of female white rhinoceroses in the preserve.

c. Realistically, not all of the rhinoceroses survive each year, so we assume a 5% death rate of all rhinoceroses. Now what type of function should be used to model the population of female white rhinoceroses in the preserve?

d. Find a function to model the population of female white rhinoceroses in the preserve, taking into account the births of new calves and the 5% death rate.


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11. The table below shows tide data for Montauk, NY, for May 21–22, 2014. The heights reported in the table are relative to the Mean Lower Low Water (MLLW). The MLLW is the average height of the lowest tide recorded at a tide station each day during the recording period. This reference point is used by the National Oceanic and Atmospheric Administration (NOAA) for the purposes of reporting tidal data throughout the United States. Each different tide station throughout the United States has its own MLLW. High and low tide levels are reported relative to this number. Since it is an average, some low tide values can be negative. NOAA resets the MLLW values approximately every 20 years.

MONTAUK, NY, TIDE CHART

Date Day Time Height in Feet High/Low 2014/05/21 Wed. 02:47 a.m. 2.48 H 2014/05/21 Wed. 09:46 a.m. −0.02 L 2014/05/21 Wed. 03:31 p.m. 2.39 H 2014/05/21 Wed. 10:20 p.m. 0.27 L 2014/05/22 Thurs. 03:50 a.m. 2.30 H 2014/05/22 Thurs. 10:41 a.m. 0.02 L 2014/05/22 Thurs. 04:35 p.m. 2.51 H 2014/05/22 Thurs. 11:23 p.m. 0.21 L

A. Create a scatter plot of the data with the horizontal axis representing time since midnight on May 21 and the vertical axis representing the height in feet relative to the MLLW.

B. What type of function would best model this set of data? Explain your choice.


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C. What are the midline, amplitude, and period of the graph?

D. Estimate the horizontal distance between a point on the graph that crosses the midline and the vertical axis.

E. Write a function of the form 𝑓𝑓(𝑥𝑥) = 𝐴𝐴 sin�𝜔𝜔(𝑥𝑥 − ℎ)� + 𝑘𝑘 to model these data, where 𝑥𝑥 is the hours since midnight on May 21, and 𝑓𝑓(𝑥𝑥) is the height in feet relative to the MLLW.

12. The graph of the tides at Montauk for the week of May 21–28 is shown below. How accurately does your model predict the time and height of the high tide on May 28?

Source: http://tidesandcurrents.noaa.gov/


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http://tidesandcurrents.noaa.gov/noaatidepredictions/NOAATidesFacade.jsp?timeZone=2&dataUnits=1&datum=MLLW&timeUnits=2&interval=highlow&Threshold=greaterthanequal&thresholdvalue=&format=Submit&Stationid=8510560&&bmon=05&bday=21&byear=2014&edate=&timelength=weekly


13. A. Stock prices have historically increased over time, but they also vary in a cyclical fashion. Create a scatter plot of the data for the monthly stock price for a 15-month time period since January 1, 2003.

B. Would a sinusoidal function be an appropriate model for these data? Explain your reasoning.

Months Since Jan. 1, 2003

Price at Close in dollars

0 20.24 1 19.42 2 18.25 3 19.13 4 19.20 5 20.91 6 20.86 7 20.04 8 20.30 9 20.54

10 21.94 11 21.87 12 21.51 13 20.65 14 19.84

02468

1012141618202224

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Stock Prices


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Lesson 17: Graphing the Tangent Function

You will need: Axes for Graph of Tangent Function

Exploratory Challenge/Exercises 1–5

1. Use your calculator to calculate each value of tan(𝑥𝑥) to two decimal places in the table for your group.

Group 1

�−𝝅𝝅𝟐𝟐

,𝝅𝝅𝟐𝟐�

Group 2

�𝝅𝝅𝟐𝟐

,𝟑𝟑𝝅𝝅𝟐𝟐�

Group 3

�−𝟑𝟑𝝅𝝅𝟐𝟐

,−𝝅𝝅𝟐𝟐�

Group 4

�𝟑𝟑𝝅𝝅𝟐𝟐

,𝟓𝟓𝝅𝝅𝟐𝟐�

𝒙𝒙 𝐭𝐭𝐭𝐭𝐭𝐭(𝒙𝒙) 𝒙𝒙 𝐭𝐭𝐭𝐭𝐭𝐭(𝒙𝒙) 𝒙𝒙 𝐭𝐭𝐭𝐭𝐭𝐭(𝒙𝒙) 𝒙𝒙 𝐭𝐭𝐭𝐭𝐭𝐭(𝒙𝒙)

−11𝜋𝜋24

13𝜋𝜋24

−35𝜋𝜋24

37𝜋𝜋24

−5𝜋𝜋12

7𝜋𝜋12

−17𝜋𝜋12

19𝜋𝜋12

−4𝜋𝜋12

8𝜋𝜋12

−16𝜋𝜋12

20𝜋𝜋12

−3𝜋𝜋12

9𝜋𝜋12

−15𝜋𝜋12

21𝜋𝜋12

−2𝜋𝜋12

10𝜋𝜋12

−14𝜋𝜋12

22𝜋𝜋12

−𝜋𝜋

12

11𝜋𝜋12

−13𝜋𝜋12

23𝜋𝜋12

0 𝜋𝜋 −𝜋𝜋 2𝜋𝜋

𝜋𝜋12

13𝜋𝜋12

−11𝜋𝜋12

25𝜋𝜋12

2𝜋𝜋12

14𝜋𝜋12

−10𝜋𝜋12

26𝜋𝜋12

3𝜋𝜋12

15𝜋𝜋12

−9𝜋𝜋12

27𝜋𝜋12

4𝜋𝜋12

16𝜋𝜋12

−8𝜋𝜋12

28𝜋𝜋12

5𝜋𝜋12

17𝜋𝜋12

−7𝜋𝜋12

29𝜋𝜋12

11𝜋𝜋24

35𝜋𝜋24

−13𝜋𝜋24

59𝜋𝜋24

Lesson 17: Graphing the Tangent Function Unit 7: Transformations & Identities

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Group 5

�−𝟓𝟓𝝅𝝅𝟐𝟐

,−𝟑𝟑𝝅𝝅𝟐𝟐�

Group 6

�𝟓𝟓𝝅𝝅𝟐𝟐

,𝟕𝟕𝝅𝝅𝟐𝟐�

Group 7

�−𝟕𝟕𝝅𝝅𝟐𝟐

,−𝟓𝟓𝝅𝝅𝟐𝟐�

Group 8

�𝟕𝟕𝝅𝝅𝟐𝟐

,𝟗𝟗𝝅𝝅𝟐𝟐�

𝒙𝒙 𝐭𝐭𝐭𝐭𝐭𝐭(𝒙𝒙) 𝒙𝒙 𝐭𝐭𝐭𝐭𝐭𝐭(𝒙𝒙) 𝒙𝒙 𝐭𝐭𝐭𝐭𝐭𝐭(𝒙𝒙) 𝒙𝒙 𝐭𝐭𝐭𝐭𝐭𝐭(𝒙𝒙)

−59𝜋𝜋24

61𝜋𝜋24

−83𝜋𝜋24

37𝜋𝜋24

−29𝜋𝜋12

31𝜋𝜋12

−41𝜋𝜋12

43𝜋𝜋12

−28𝜋𝜋12

32𝜋𝜋12

−40𝜋𝜋12

44𝜋𝜋12

−27𝜋𝜋12

33𝜋𝜋12

−39𝜋𝜋12

45𝜋𝜋12

−26𝜋𝜋12

34𝜋𝜋12

−38𝜋𝜋12

46𝜋𝜋12

−25𝜋𝜋12

35𝜋𝜋12

−37𝜋𝜋12

47𝜋𝜋12

−2𝜋𝜋 3𝜋𝜋 −3𝜋𝜋 4𝜋𝜋

−23𝜋𝜋12

37𝜋𝜋12

−35𝜋𝜋12

49𝜋𝜋12

−22𝜋𝜋12

38𝜋𝜋12

−34𝜋𝜋12

50𝜋𝜋12

−21𝜋𝜋12

39𝜋𝜋12

−33𝜋𝜋12

51𝜋𝜋12

−20𝜋𝜋12

40𝜋𝜋12

−32𝜋𝜋12

52𝜋𝜋12

−19𝜋𝜋12

41𝜋𝜋12

−31𝜋𝜋12

53𝜋𝜋12

−37𝜋𝜋24

83𝜋𝜋24

−61𝜋𝜋24

107𝜋𝜋

24

2. The tick marks on the axes provided are spaced in increments of 𝜋𝜋12

. Mark the horizontal axis by writing the number of the left endpoint of your interval at the leftmost tick mark, the multiple of 𝜋𝜋 that is in the middle of your interval at the point where the axes cross, and the number at the right endpoint of your interval at the rightmost tick mark. Fill in the remaining values at increments of 𝜋𝜋

12.


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3. On your plot, sketch the graph of 𝑦𝑦 = tan(𝑥𝑥) on your specified interval by plotting the points in the table and connecting the points with a smooth curve. Draw the graph with a bold marker.

4. What happens to the graph near the edges of your interval? Why does this happen?

5. When you are finished, affix your graph to the board in the appropriate place, matching endpoints of intervals.

The breaks in the graph are examples of vertical asymptotes. These lines that the graph gets very close to but does not cross often occur at a value 𝑥𝑥 = 𝑎𝑎, where the function is undefined to prevent division by zero.

Exploratory Challenge 2/Exercises 6–16

For each exercise below, let 𝑚𝑚 = tan(𝜃𝜃) be the slope of the terminal ray in the definition of the tangent function, and let 𝑃𝑃 = (𝑥𝑥0,𝑦𝑦0) be the intersection of the terminal ray with the unit circle after being rotated by 𝜃𝜃 radians for 0 < 𝜃𝜃 < 𝜋𝜋

2. We know that the tangent of 𝜃𝜃 is the slope 𝑚𝑚 of 𝑂𝑂𝑃𝑃�⃖��⃗ .

6. Let 𝑄𝑄 be the intersection of the terminal ray with the unit circle after being rotated by 𝜃𝜃 + 𝜋𝜋 radians.

A. What is the slope of 𝑂𝑂𝑄𝑄�⃖��⃗ ?

B. Find an expression for tan(𝜃𝜃 + 𝜋𝜋) in terms of 𝑚𝑚.

C. Find an expression for tan(𝜃𝜃 + 𝜋𝜋) in terms of tan(𝜃𝜃).

D. How can the expression in Part C be seen in the graph of the tangent function?


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7. Let 𝑄𝑄 be the intersection of the terminal ray with the unit circle after being rotated by −𝜃𝜃 radians.


B. Find an expression for tan(−𝜃𝜃) in terms of 𝑚𝑚.

C. Find an expression for tan(−𝜃𝜃) in terms of tan(𝜃𝜃).

D. How can the expression in Part C be seen in the graph of the tangent function?

8. Is the tangent function an even function, an odd function, or neither? How can you tell your answer is correct from the graph of the tangent function?

9. Let 𝑄𝑄 be the intersection of the terminal ray with the unit circle after being rotated by 𝜋𝜋 − 𝜃𝜃 radians.


B. Find an expression for tan(𝜋𝜋 − 𝜃𝜃) in terms of 𝑚𝑚.

C. Find an expression for tan(𝜋𝜋 − 𝜃𝜃) in terms of tan(𝜃𝜃).


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10. Let 𝑄𝑄 be the intersection of the terminal ray with the unit circle after being rotated by 𝜋𝜋2

+ 𝜃𝜃 radians.


B. Find an expression for tan �𝜋𝜋2

+ 𝜃𝜃� in terms of 𝑚𝑚.

C. Find an expression for tan �𝜋𝜋2

+ 𝜃𝜃 � first in terms of tan(𝜃𝜃) and then in terms of cot(𝜃𝜃).

11. Let 𝑄𝑄 be the intersection of the terminal ray with the unit circle after being rotated by 𝜋𝜋2− 𝜃𝜃 radians.


B. Find an expression for tan �𝜋𝜋2− 𝜃𝜃� in terms of

𝑚𝑚.

C. Find an expression for tan �𝜋𝜋2− 𝜃𝜃� in terms of

tan(𝜃𝜃) or other trigonometric functions. 12. Summarize your results from Exercises 6, 7, 9, 10, and 11.


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13. We have only demonstrated that the identities in Exercise 12 are valid for 0 < θ < 𝜋𝜋2

because we only

used rotations that left point 𝑃𝑃 in the first quadrant. Argue that tan �− 2𝜋𝜋3� = −tan �2𝜋𝜋

3�. Then, using

similar logic, we could argue that all of the above identities extend to any value of 𝜃𝜃 for which the tangent (and cotangent for the last two) is defined.

14. For which values of 𝜃𝜃 are the identities in Exercise 12 valid? 15. Derive an identity for tan(2𝜋𝜋 + 𝜃𝜃) from the graph.

16. Use the identities you summarized in Exercise 12 to show tan(2𝜋𝜋 − 𝜃𝜃) = −tan(𝜃𝜃) where 𝜃𝜃 ≠ 𝜋𝜋2

+ 𝑘𝑘𝜋𝜋, for all integers 𝑘𝑘.


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Lesson Summary

The tangent function tan(𝑥𝑥) = sin(𝑥𝑥)cos(𝑥𝑥) is periodic with period 𝜋𝜋. The following identities have

been established.

tan(𝑥𝑥 + 𝜋𝜋) = tan(𝑥𝑥) for all 𝑥𝑥 ≠ 𝜋𝜋2


tan(−𝑥𝑥) = −tan(𝑥𝑥) for all 𝑥𝑥 ≠ 𝜋𝜋2


tan(𝜋𝜋 − 𝑥𝑥) = −tan(𝑥𝑥) for all 𝑥𝑥 ≠ 𝜋𝜋2


tan �𝜋𝜋2

+ 𝑥𝑥� = −cot(𝑥𝑥) for all 𝑥𝑥 ≠ 𝑘𝑘𝜋𝜋, for all integers 𝑘𝑘.

tan �𝜋𝜋2− 𝑥𝑥� = cot(𝑥𝑥) for all 𝑥𝑥 ≠ 𝑘𝑘𝜋𝜋, for all integers 𝑘𝑘.

tan(2𝜋𝜋 + 𝑥𝑥) = tan(𝑥𝑥) for all 𝑥𝑥 ≠ 𝜋𝜋2


tan(2𝜋𝜋 − 𝑥𝑥) = −tan(𝑥𝑥) for all 𝑥𝑥 ≠ 𝜋𝜋2



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Homework Problem Set

1. Recall that the cotangent function is defined by cot(𝑥𝑥) = cos(𝑥𝑥)sin(𝑥𝑥)

= 1tan(𝑥𝑥)

, where sin(𝑥𝑥) ≠ 0.

a. What is the domain of the cotangent function? Explain how you know.

b. What is the period of the cotangent function? Explain how you know.

c. Use a calculator to complete the table of values of the cotangent function on the interval (0,𝜋𝜋) to two decimal places.

𝒙𝒙 𝐜𝐜𝐜𝐜𝐭𝐭(𝒙𝒙) 𝒙𝒙 𝐜𝐜𝐜𝐜𝐭𝐭(𝒙𝒙) 𝒙𝒙 𝐜𝐜𝐜𝐜𝐭𝐭(𝒙𝒙) 𝒙𝒙 𝐜𝐜𝐜𝐜𝐭𝐭(𝒙𝒙)

𝜋𝜋24

4𝜋𝜋12

7𝜋𝜋12

10𝜋𝜋12

𝜋𝜋12

5𝜋𝜋12

8𝜋𝜋12

11𝜋𝜋12

2𝜋𝜋12

𝜋𝜋2

9𝜋𝜋12

23𝜋𝜋24

3𝜋𝜋12

d. Plot your data from part (c), and sketch a graph of 𝑦𝑦 = cot(𝑥𝑥) on (0,𝜋𝜋).


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e. Sketch a graph of 𝑦𝑦 = cot(𝑥𝑥) on (−2𝜋𝜋, 2𝜋𝜋) without plotting points.

f. Discuss the similarities and differences between the graphs of the tangent and cotangent functions.

g. Find all 𝑥𝑥-values where tan(𝑥𝑥) = cot(𝑥𝑥) on the interval (0, 2𝜋𝜋).


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2. Each set of axes below shows the graph of 𝑓𝑓(𝑥𝑥) = tan(𝑥𝑥). Use what you know about function transformations to sketch a graph of 𝑦𝑦 = 𝑔𝑔(𝑥𝑥) for each function 𝑔𝑔 on the interval (0, 2𝜋𝜋). a. 𝑔𝑔(𝑥𝑥) = 2 tan(𝑥𝑥)

b. 𝑔𝑔(𝑥𝑥) = 13 tan(𝑥𝑥)


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c. 𝑔𝑔(𝑥𝑥) = −2 tan(𝑥𝑥)

d. How does changing the parameter 𝐴𝐴 affect the graph of 𝑔𝑔(𝑥𝑥) = 𝐴𝐴 tan(𝑥𝑥)?

3. Each set of axes below shows the graph of 𝑓𝑓(𝑥𝑥) = tan(𝑥𝑥). Use what you know about function transformations to sketch a graph of 𝑦𝑦 = 𝑔𝑔(𝑥𝑥) for each function 𝑔𝑔 on the interval (0, 2𝜋𝜋).

a. 𝑔𝑔(𝑥𝑥) = tan �𝑥𝑥 − 𝜋𝜋2�


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b. 𝑔𝑔(𝑥𝑥) = tan �𝑥𝑥 − 𝜋𝜋6�

c. 𝑔𝑔(𝑥𝑥) = tan �𝑥𝑥 + 𝜋𝜋4�

d. How does changing the parameter ℎ affect the graph of 𝑔𝑔(𝑥𝑥) = tan(𝑥𝑥 − ℎ)?


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4. Each set of axes below shows the graph of 𝑓𝑓(𝑥𝑥) = tan(𝑥𝑥). Use what you know about function transformations to sketch a graph of 𝑦𝑦 = 𝑔𝑔(𝑥𝑥) for each function 𝑔𝑔 on the interval (0, 2𝜋𝜋). a. 𝑔𝑔(𝑥𝑥) = tan(𝑥𝑥) + 1

b. 𝑔𝑔(𝑥𝑥) = tan(𝑥𝑥) + 3


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c. 𝑔𝑔(𝑥𝑥) = tan(𝑥𝑥) − 2

d. How does changing the parameter 𝑘𝑘 affect the graph of 𝑔𝑔(𝑥𝑥) = tan(𝑥𝑥) + 𝑘𝑘?

5. Each set of axes below shows the graph of 𝑓𝑓(𝑥𝑥) = tan(𝑥𝑥). Use what you know about function transformations to sketch a graph of 𝑦𝑦 = 𝑔𝑔(𝑥𝑥) for each function 𝑔𝑔 on the interval (0, 2𝜋𝜋). a. 𝑔𝑔(𝑥𝑥) = tan(3𝑥𝑥)


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b. 𝑔𝑔(𝑥𝑥) = tan �𝑥𝑥2�

c. 𝑔𝑔(𝑥𝑥) = tan(−3𝑥𝑥)

d. How does changing the parameter 𝜔𝜔 affect the graph of 𝑔𝑔(𝑥𝑥) = tan(𝜔𝜔𝑥𝑥)?


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6. Use your knowledge of function transformation and the graph of 𝑦𝑦 = tan(𝑥𝑥) to sketch graphs of the following

transformations of the tangent function. a. 𝑦𝑦 = tan(2𝑥𝑥)

b. 𝑦𝑦 = tan �2 �𝑥𝑥 − 𝜋𝜋4��


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c. 𝑦𝑦 = tan �2 �𝑥𝑥 − 𝜋𝜋4�� + 1.5

7. Find parameters 𝐴𝐴, 𝜔𝜔, ℎ, and 𝑘𝑘 so that the graphs of 𝑓𝑓(𝑥𝑥) = 𝐴𝐴 tan�𝜔𝜔(𝑥𝑥 − ℎ)� + 𝑘𝑘 and 𝑔𝑔(𝑥𝑥) = cot(𝑥𝑥) are the same.


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S.176









Lesson 18: What Is a Trigonometric Identity?

Opening Exercise: The Pythagorean Identity

Lessons 5 and 6 extended the definitions of the sine and cosine functions so that sin(𝜃𝜃) and cos(𝜃𝜃) are defined for all real numbers 𝜃𝜃.

1. What is the equation of the unit circle centered at the origin?

2. Recall that this equation is a special case of the Pythagorean Theorem. Given the number 𝜃𝜃, there is a unique point 𝑃𝑃 on the unit circle that results from rotating the positive 𝑥𝑥-axis through 𝜃𝜃 radians around the origin. What are the coordinates of 𝑃𝑃?

3. How can you combine this information to get a formula involving sin(𝜃𝜃) and cos(𝜃𝜃)?

Lesson 18: What Is a Trigonometric Identity? Unit 7: Transformations & Identities

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The equation sin2(𝜃𝜃) + cos2(𝜃𝜃) = 1 is true for all real numbers 𝜃𝜃 and is an identity.

The functions on either side of the equal sign are equivalent for every value of 𝜃𝜃. They have the same domain, the same range, and the same rule of assignment. You saw some polynomial identities in Module 1, and we’ve developed some identities for sine, cosine, and tangent observed from graphs in this module. The identity we just proved is a trigonometric identity, and it is called the Pythagorean identity because it is another important consequence of the Pythagorean theorem. Notice that we use the notation sin2(𝜃𝜃) in place of (sin(𝜃𝜃))2. Both are correct, but the first is notationally simpler. Notice also that neither is the same expression as sin(𝜃𝜃2).

Another Identity?

4. A. Divide both sides of the Pythagorean identity by cos2(𝜃𝜃). What happens to the identity?

B. What happens when 𝜃𝜃 = −𝜋𝜋2

, 𝜃𝜃 = 𝜋𝜋2

, and 𝜃𝜃 = 3𝜋𝜋2

? Why?

C. How do we need to modify our claim about what looks like a new identity? D. For which values of 𝜃𝜃 are the functions 𝑓𝑓(𝜃𝜃) = tan2(𝜃𝜃) + 1 and 𝑔𝑔(𝜃𝜃) = sec2(𝜃𝜃) defined? E. What is the range of each of the functions 𝑓𝑓(𝜃𝜃) = tan2(𝜃𝜃) + 1 and 𝑔𝑔(𝜃𝜃) = sec2(𝜃𝜃)?


S.178









The two functions have the same domain and the same range, and they are equivalent.

Therefore, tan2(𝜃𝜃) + 1 = sec2(𝜃𝜃) is a trigonometric identity for all real

numbers 𝜃𝜃 such that 𝜃𝜃 ≠ 𝜋𝜋2

+ 𝑘𝑘𝑘𝑘, for all integers 𝑘𝑘.

For any trigonometric identity, we need to specify not only the two functions that are equivalent, but also the values for which the identity is true.

Using the Pythagorean Identities

Recall the Pythagorean identity sin2(𝜃𝜃) + cos2(𝜃𝜃) = 1, where 𝜃𝜃 is any real number.

5. Find sin(𝑥𝑥), given cos(𝑥𝑥) = 35, for −𝑘𝑘

2 < 𝑥𝑥 < 0.

6. Find tan(𝑦𝑦), given cos(𝑦𝑦) = − 513, for

𝜋𝜋2

< 𝑦𝑦 < 𝑘𝑘.

7. Write tan(𝑧𝑧) in terms of cos(𝑧𝑧), for 𝑘𝑘 < 𝑧𝑧 < 3𝑘𝑘2 .


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8. Use the Pythagorean identity to do the following:

A. Rewrite the expression cos(𝜃𝜃) sin2(𝜃𝜃)− cos(𝜃𝜃) in terms of a single trigonometric function. State the resulting identity.

B. Rewrite the expression (1 − cos2(𝜃𝜃)) csc(𝜃𝜃) in terms of a single trigonometric function. State the resulting identity.

C. Find all solutions to the equation 2 sin2(𝜃𝜃) = 2 + cos(𝜃𝜃) in the interval (0, 2𝑘𝑘). Draw a unit circle that shows the solutions.


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9. Which of the following statements are identities? If a statement is an identity, specify the values of x where the equation holds.

A. sin(𝑥𝑥 + 2𝑘𝑘) = sin(𝑥𝑥) where the functions on both sides are defined.

B. sec(𝑥𝑥) = 1 where the functions on both sides are defined.

C. sin(−𝑥𝑥) = sin(𝑥𝑥) where the functions on both sides are defined.

D. 1 + tan2(𝑥𝑥) = sec2(𝑥𝑥) where the functions on both sides are defined.

E. sin �𝑘𝑘2 − 𝑥𝑥� = cos(𝑥𝑥) where the functions on both sides are defined.

F. sin2(𝑥𝑥) = tan2(𝑥𝑥) for all real 𝑥𝑥.


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Lesson Summary

The Pythagorean identities: 𝐬𝐬𝐬𝐬𝐬𝐬𝟐𝟐(𝜽𝜽) + 𝐜𝐜𝐜𝐜𝐬𝐬𝟐𝟐(𝜽𝜽) = 𝟏𝟏 for all real numbers 𝜽𝜽 𝐭𝐭𝐭𝐭𝐬𝐬𝟐𝟐(𝜽𝜽) + 𝟏𝟏 = 𝐬𝐬𝐬𝐬𝐜𝐜𝟐𝟐(𝜽𝜽) for all real numbers 𝜽𝜽 such that 𝜽𝜽 ≠ 𝝅𝝅

𝟐𝟐+ 𝒌𝒌𝝅𝝅, for all integers 𝒌𝒌

𝟏𝟏 + 𝐜𝐜𝐜𝐜𝐭𝐭𝟐𝟐(𝜽𝜽) = 𝐜𝐜𝐬𝐬𝐜𝐜𝟐𝟐(𝜽𝜽) for all real numbers 𝜽𝜽 such that 𝛉𝛉 ≠ 𝐤𝐤𝐤𝐤, for all integers 𝐤𝐤.


S.182









Homework Problem Set 1. Which of the following statements are trigonometric identities? Graph the functions on each side of the equation

on a graphing utility.

A. tan(𝑥𝑥) = sin(𝑥𝑥)cos(𝑥𝑥)

where the functions on both sides are defined.

B. cos2(𝑥𝑥) = 1 + sin(𝑥𝑥) where the functions on both sides are defined.

C. cos �𝑘𝑘2 − 𝑥𝑥� = sin(𝑥𝑥) where the functions on both sides are defined.

2. Determine the domain of the following trigonometric identities:

A. cot(𝑥𝑥) = cos(𝑥𝑥)sin(𝑥𝑥)


B. cos(−𝑢𝑢) = cos(𝑢𝑢) where the functions on both sides are defined.

C. sec(𝑦𝑦) = 1cos(𝑦𝑦)


3. Rewrite sin(𝑥𝑥)cos2(𝑥𝑥) − sin(𝑥𝑥) as an expression containing a single term.

4. Suppose 0 < 𝜃𝜃 < 𝑘𝑘2 and sin(𝜃𝜃) = 1

�3. What is the value of cos(𝜃𝜃)?


S.183









5. If cos(𝜃𝜃) = − 1�5

, what are possible values of sin(𝜃𝜃)?

6. Use the Pythagorean identity sin2(𝜃𝜃) + cos2(𝜃𝜃) = 1, where 𝜃𝜃 is any real number, to find the following:

A. cos(𝜃𝜃), given sin(𝜃𝜃) = 513, for

𝜋𝜋2

< 𝜃𝜃 < 𝑘𝑘.

B. tan(𝑥𝑥), given cos(𝑥𝑥) = − 1�2

, for 𝑘𝑘 < 𝑥𝑥 < 3𝑘𝑘2 .

7. The three identities below are all called Pythagorean identities. The second and third follow from the first, as you saw in Example 1 and the Exit Ticket.

A. For which values of θ are each of these identities defined?

i. sin2(𝜃𝜃) + cos2(𝜃𝜃) = 1, where the functions on both sides are defined.

ii. tan2(𝜃𝜃) + 1 = sec2(𝜃𝜃), where the functions on both sides are defined.

iii. 1 + cot2(𝜃𝜃) = csc2(𝜃𝜃), where the functions on both sides are defined.

B. For which of the three identities is 0 in the domain of validity?

C. For which of the three identities is 𝜋𝜋2

in the domain of validity?

D. For which of the three identities is −𝑘𝑘4 in the domain of validity?


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Lesson 19: Proving Trigonometric Identities

Opening Exercise

Over the course of this module you’ve learned 11 trigonometric identities. Let’s see if you can remember them all!

1. In the first column is the name of the identity and the second column contains the one side of the trig identity. Write in the rest of the trig identity to make the statement true.

Reci

proc

al I

dent

itie

s

sinθ =

cosθ =

tanθ =

cscθ =

secθ =

cotθ =

Quo

tien

t Id

enti

ties

tanθ =

cotθ =

Pyth

agor

ean

Iden

titi

es sin2θ + cos2θ =

1 + cot2θ =

tan2θ + 1 =

Lesson 19: Proving Trigonometric Identities Unit 7: Transformations & Identities

S.185









2. Which of these statements is a trigonometric identity? Provide evidence to support your claim.

Statement 1: sin2(𝜃𝜃) = 1 − cos2(𝜃𝜃) for 𝜃𝜃 any real number.

Statement 2: 1 − cos(𝜃𝜃) = 1 − cos(𝜃𝜃) for 𝜃𝜃 any real number.

Statement 3: 1 − cos(𝜃𝜃) = 1 + cos(𝜃𝜃) for 𝜃𝜃 any real number.

3. Discuss what it might mean to “prove” an identity. What might it take to prove, for example, that the following statement is an identity?

sin2(𝜃𝜃)1−cos(𝜃𝜃)

= 1 + cos(𝜃𝜃) where 𝜃𝜃 ≠ 2𝜋𝜋𝜋𝜋, for all integers 𝜋𝜋.


S.186









To prove an identity, you have to use logical steps to show that one side of the equation in the identity can be transformed into the other side of the equation using already established

identities such as the Pythagorean identity or the properties of operation (commutative, associative, and distributive properties).

4. Below are properties you saw in Algebra 1 and Geometry. We’ll make use of these in our proofs of trigonometric identities. Complete each one.

Reflexive property of equality

𝑎𝑎 = _____

Symmetric property of equality

If 𝑎𝑎 = 𝑏𝑏, then 𝑏𝑏 = _____.

Transitive property of equality

If 𝑎𝑎 = 𝑏𝑏 and 𝑏𝑏 = 𝑐𝑐, then 𝑎𝑎 = _____.

Addition property of equality

If 𝑎𝑎 = 𝑏𝑏, then 𝑎𝑎 + 𝑐𝑐 = 𝑏𝑏 + _____.

Subtraction property of equality

If 𝑎𝑎 = 𝑏𝑏, then 𝑎𝑎 − 𝑐𝑐 = _____ − _____.

Multiplication property of equality

If 𝑎𝑎 = 𝑏𝑏, then 𝑎𝑎 ⋅ 𝑐𝑐 = 𝑏𝑏 ⋅ _____.

Division property of equality

If 𝑎𝑎 = 𝑏𝑏 and 𝑐𝑐 ≠ 0, then 𝑎𝑎 ÷ 𝑐𝑐 = _____ ÷ 𝑐𝑐.

Substitution property of equality

If 𝑎𝑎 = 𝑏𝑏, then 𝑏𝑏 may be substituted for _____ in any expression containing 𝑎𝑎.

Associative properties (𝑎𝑎 + 𝑏𝑏) + 𝑐𝑐 = 𝑎𝑎 + (_____ + 𝑐𝑐) and 𝑎𝑎(𝑏𝑏𝑐𝑐) = (_______)𝑐𝑐.

Commutative properties 𝑎𝑎 + 𝑏𝑏 = _____ + 𝑎𝑎 and 𝑎𝑎𝑏𝑏 = 𝑏𝑏_____.

Distributive property 𝑎𝑎(𝑏𝑏 + 𝑐𝑐) = _______ + 𝑎𝑎𝑐𝑐 and (𝑎𝑎 + 𝑏𝑏)𝑐𝑐 = 𝑎𝑎𝑐𝑐 + _______.


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Two Proofs of Our New Identity

Work through these two different ways to approach proving the identity sin2(𝜃𝜃)

1−cos(𝜃𝜃) = 1 + cos(𝜃𝜃) where 𝜃𝜃 ≠ 2𝜋𝜋𝜋𝜋,

for integers 𝜋𝜋. Here 𝑎𝑎, 𝑏𝑏, and 𝑐𝑐 stand for arbitrary real numbers. In the first proof (Exercise 4A), we start with an identity and use mathematical rules to get other true statements. In the second proof (Exercise 4B), we start with the statement we want to prove and then only work with one side of that statement to show that the two sides are identical.

5. Fill in the missing parts of the proofs outlined in the tables below.

A. We start with the Pythagorean identity. When we divide both sides by the same expression, 1 − cos(𝜃𝜃), we introduce potential division by zero when cos(𝜃𝜃) = 1. This will change the set of values of 𝜃𝜃 for which the identity is valid.

PROOF:

Step Left Side of Equation

Equivalent Right Side Domain Reason

1 sin2(𝜃𝜃) + cos2(𝜃𝜃) = 1 𝜃𝜃 any real number Pythagorean identity

2 sin2(𝜃𝜃) = 1 − cos2(𝜃𝜃) 𝜃𝜃 any real number

3 = �1 − cos(𝜃𝜃)��1 + cos(𝜃𝜃)� 𝜃𝜃 any real number

4 = �1 − cos(𝜃𝜃)��1 + cos(𝜃𝜃)�

1 − cos(𝜃𝜃)

5 sin2(𝜃𝜃)

1 − cos(𝜃𝜃) = 𝜃𝜃 ≠ 2𝜋𝜋𝜋𝜋, for all

integers 𝜋𝜋

Substitution property of

equality using 1−cos(𝜃𝜃)1−cos(𝜃𝜃) =

1


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B. Or, we can start with the more complicated side of the identity we want to prove and use algebra and prior trigonometric definitions and identities to transform it to the other side. In this case, the more

complicated expression is sin2(𝜃𝜃)

1−cos(𝜃𝜃).

PROOF:

Step Left Side of

Equation Equivalent Right Side Domain Reason

1 sin2(𝜃𝜃)

1 − cos(𝜃𝜃) = 1 − cos2(𝜃𝜃)1 − cos(𝜃𝜃)

𝜃𝜃 ≠ 2𝜋𝜋𝜋𝜋, for all integers 𝜋𝜋

Substitution property of equality using

sin2(𝜃𝜃) = 1 − cos2(𝜃𝜃)

2 = �1 − cos(𝜃𝜃)��1 + cos(𝜃𝜃)�

1 − cos (𝜃𝜃) Distributive property

3 sin2(𝜃𝜃)

1 − cos(𝜃𝜃) = 1 + cos(𝜃𝜃)

We will mostly use the second method for our proofs.

Practice Exercises 6–7

Prove that the following are trigonometric identities, beginning with the side of the equation that seems to be more complicated and starting the proof by restricting 𝑥𝑥 to values where the identity is valid. Make sure that the complete identity statement is included at the end of the proof.

6. tan(𝑥𝑥) = sec(𝑥𝑥)csc(𝑥𝑥) for real numbers 𝑥𝑥 ≠ 𝜋𝜋

2+ 𝜋𝜋𝜋𝜋, for all integers 𝜋𝜋.


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7. cot(𝑥𝑥) + tan(𝑥𝑥) = sec(𝑥𝑥) csc(𝑥𝑥) for all real numbers 𝑥𝑥 ≠ 𝜋𝜋2𝑛𝑛 for integer 𝑛𝑛.


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Homework Problem Set 1. Does sin(𝑥𝑥 + 𝑦𝑦) equal sin(𝑥𝑥) + sin(𝑦𝑦) for all real numbers 𝑥𝑥 and 𝑦𝑦?

a. Find each of the following: sin �𝜋𝜋2� , sin �𝜋𝜋

4� , sin �3𝜋𝜋

4�.

b. Are sin �𝜋𝜋2

+ 𝜋𝜋4� and sin �𝜋𝜋

2� + sin �𝜋𝜋

4� equal?

c. Are there any values of 𝑥𝑥 and 𝑦𝑦 for which sin(𝑥𝑥 + 𝑦𝑦) = sin(𝑥𝑥) + sin(𝑦𝑦)?

2. Use tan(𝑥𝑥) = sin(𝑥𝑥)cos(𝑥𝑥) and identities involving the sine and cosine functions to establish the following

identities for the tangent function. Identify the values of 𝑥𝑥 where the equation is an identity. a. tan(𝜋𝜋 − 𝑥𝑥) = tan(𝑥𝑥)


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b. tan(𝑥𝑥 + 𝜋𝜋) = tan(𝑥𝑥)

c. tan(2𝜋𝜋 − 𝑥𝑥) = −tan(𝑥𝑥)

d. tan(−𝑥𝑥) = −tan(𝑥𝑥)


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3. Rewrite each of the following expressions as a single term. Identify the values of 𝑥𝑥 for which the original expression and your expression are equal: a. cot(𝑥𝑥) sec(𝑥𝑥) sin(𝑥𝑥)

b. � 11−sin(𝑥𝑥)� �

11+sin(𝑥𝑥)�

c. 1cos2(𝑥𝑥) −

1cot2(𝑥𝑥)


S.193









d. �tan(𝑥𝑥)−sin(𝑥𝑥)��1+cos(𝑥𝑥)�sin3(𝑥𝑥)

4. Prove that for any two real numbers 𝑎𝑎 and 𝑏𝑏, sin2(𝑎𝑎) − sin2(𝑏𝑏) + cos2(𝑎𝑎) sin2(𝑏𝑏) − sin2(𝑎𝑎) cos2(𝑏𝑏) = 0.


S.194









5. Prove that the following statements are identities for all values of 𝜃𝜃 for which both sides are defined, and describe that set. a. cot(𝜃𝜃)sec(𝜃𝜃) = csc(𝜃𝜃)

b. �csc(𝜃𝜃) + cot(𝜃𝜃)��1 − cos(𝜃𝜃)� = sin(𝜃𝜃)

c. tan2(𝜃𝜃) − sin2(𝜃𝜃) = tan2(𝜃𝜃) sin2(𝜃𝜃)


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d. 4+tan2(𝑥𝑥)−sec2(𝑥𝑥)csc2(𝑥𝑥) = 3 sin2(𝑥𝑥)

e. �1+sin(𝜃𝜃)�2+ cos2(𝜃𝜃)

1+sin(𝜃𝜃) = 2


S.196









6. Prove that the value of the following expression does not depend on the value of 𝑦𝑦:

cot(𝑦𝑦)tan(𝑥𝑥) + tan(𝑦𝑦)cot(𝑥𝑥) + cot(𝑦𝑦).


S.197










S.198







HONORS ALGEBRA I/TRIGONOMETRYI

Hart Interactive – Honors Algebra 2/Trig M3 Lesson 20

Lesson 20: Trigonometric Identity Proofs

Exploratory Activity

1. We have seen that sin(𝛼𝛼 + 𝛽𝛽) ≠ sin(𝛼𝛼) + sin(𝛽𝛽). So, what is sin(𝛼𝛼 + 𝛽𝛽)?

A. Begin by completing the following table:

𝜶𝜶 𝜷𝜷 𝐬𝐬𝐬𝐬𝐬𝐬(𝜶𝜶) 𝐬𝐬𝐬𝐬𝐬𝐬(𝜷𝜷) 𝐬𝐬𝐬𝐬𝐬𝐬(𝜶𝜶 + 𝜷𝜷) 𝐬𝐬𝐬𝐬𝐬𝐬(𝜶𝜶) 𝐜𝐜𝐜𝐜𝐬𝐬(𝜷𝜷) 𝐬𝐬𝐬𝐬𝐬𝐬(𝜶𝜶) 𝐬𝐬𝐬𝐬𝐬𝐬(𝜷𝜷) 𝐜𝐜𝐜𝐜𝐬𝐬(𝜶𝜶) 𝐜𝐜𝐜𝐜𝐬𝐬(𝜷𝜷) 𝐜𝐜𝐜𝐜𝐬𝐬(𝜶𝜶) 𝐬𝐬𝐬𝐬𝐬𝐬(𝜷𝜷)

𝜋𝜋6

𝜋𝜋6

12

12

√𝟑𝟑𝟐𝟐

√34

12

𝜋𝜋6

𝜋𝜋3

12

√32

𝟏𝟏 √34

𝜋𝜋4

𝜋𝜋6

√22

12

√𝟐𝟐 + √𝟔𝟔𝟒𝟒

√64

𝜋𝜋4

𝜋𝜋4

√22

√22

𝟏𝟏 12

12

𝜋𝜋3

𝜋𝜋3

√32

√32

√𝟑𝟑𝟐𝟐

√34

𝜋𝜋3

𝜋𝜋4

√32

√22

√𝟐𝟐 + √𝟔𝟔

𝟒𝟒 √6

4

√24

B. What pattern do you see?

sin(𝛼𝛼 + 𝛽𝛽) = _________________________________________

Lesson 20: Trigonometric Identity Proofs Unit 7: Transformations & Identities

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2. Use the following table to formulate a conjecture for cos(𝛼𝛼 + 𝛽𝛽):

𝜶𝜶 𝜷𝜷 𝐜𝐜𝐜𝐜𝐬𝐬(𝜶𝜶) 𝐜𝐜𝐜𝐜𝐬𝐬(𝜷𝜷) 𝐜𝐜𝐜𝐜𝐬𝐬(𝜶𝜶+ 𝜷𝜷) 𝐬𝐬𝐬𝐬𝐬𝐬(𝜶𝜶) 𝐜𝐜𝐜𝐜𝐬𝐬(𝜷𝜷) 𝐬𝐬𝐬𝐬𝐬𝐬(𝜶𝜶) 𝐬𝐬𝐬𝐬𝐬𝐬(𝜷𝜷) 𝐜𝐜𝐜𝐜𝐬𝐬(𝜶𝜶) 𝐜𝐜𝐜𝐜𝐬𝐬(𝜷𝜷) 𝐜𝐜𝐜𝐜𝐬𝐬(𝜶𝜶) 𝐬𝐬𝐬𝐬𝐬𝐬(𝜷𝜷)

𝜋𝜋6

𝜋𝜋6

12

12

𝟏𝟏𝟐𝟐

√34

12

34

√34

𝜋𝜋6

𝜋𝜋3

√32

12

𝟎𝟎 14

√34

√34

34

𝜋𝜋4

𝜋𝜋6

√22

√32

√𝟔𝟔 − √𝟐𝟐

𝟒𝟒

√64

√24

√64

√24

𝜋𝜋4

𝜋𝜋4

√22

√22

𝟎𝟎 12

12

12

12

𝜋𝜋3

𝜋𝜋3

12

12

−𝟏𝟏𝟐𝟐

√34

34

14

√34

𝜋𝜋3

𝜋𝜋4

12

√22

√𝟐𝟐 − √𝟔𝟔

𝟒𝟒

√64

√64

√24

√24

What pattern do you see?

cos(𝛼𝛼 + 𝛽𝛽) = _________________________________________


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Exercises 3–4: Formulas for sin(𝛼𝛼 + 𝛽𝛽) and cos(𝛼𝛼 + 𝛽𝛽)

3. The formula for the sine of the sum of two numbers is sin(𝛼𝛼 + 𝛽𝛽) = sin(𝛼𝛼) cos(𝛽𝛽) + cos(𝛼𝛼) sin(𝛽𝛽). The proof can be a little long, but it is fairly straightforward. We will prove only the case when the two numbers are positive, and their sum is less than 𝜋𝜋

2.

a. Let 𝛼𝛼 and 𝛽𝛽 be positive real numbers such that 0 < 𝛼𝛼 + 𝛽𝛽 < 𝜋𝜋2

.

b. Construct rectangle 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 such that 𝑀𝑀𝑃𝑃 = 1, 𝑚𝑚∠𝑀𝑀𝑃𝑃𝑃𝑃 = 90°,

𝑚𝑚∠𝑃𝑃𝑀𝑀𝑃𝑃 = 𝛽𝛽, and 𝑚𝑚∠𝑃𝑃𝑀𝑀𝑀𝑀 = 𝛼𝛼. See the figure on the right.

c. Fill in the blanks in terms of 𝛼𝛼 and 𝛽𝛽:

i. 𝑚𝑚∠𝑃𝑃𝑀𝑀𝑀𝑀 = .

ii. 𝑚𝑚∠𝑀𝑀𝑃𝑃𝑀𝑀 = .

iii. Therefore, sin(𝛼𝛼 + 𝛽𝛽) = 𝑀𝑀𝑀𝑀.

iv. 𝑃𝑃𝑃𝑃 = sin(______).

v. 𝑀𝑀𝑃𝑃 = cos(______).

d. Let’s label the angle and length measurements as shown.

e. Use this new figure to fill in the blanks in terms of 𝛼𝛼 and 𝛽𝛽:

i. Why does sin(𝛼𝛼) = 𝑀𝑀𝑀𝑀cos(𝛽𝛽)?

ii. Therefore, 𝑀𝑀𝑃𝑃 = _________.

iii. 𝑚𝑚∠𝑃𝑃𝑃𝑃𝑀𝑀 = ____________.

Now, consider △ 𝑃𝑃𝑃𝑃𝑀𝑀. Since cos(𝛼𝛼) = 𝑀𝑀𝑄𝑄sin(𝛽𝛽)

, 𝑃𝑃𝑀𝑀 = .

f. Label these lengths and angle measurements in the figure. g. Since 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 is a rectangle, 𝑀𝑀𝑀𝑀 = 𝑀𝑀𝑃𝑃 + 𝑃𝑃𝑀𝑀. h. Thus, sin(𝛼𝛼 + 𝛽𝛽) = sin(𝛼𝛼) cos(𝛽𝛽) + cos(𝛼𝛼) sin(𝛽𝛽).


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Note that we have only proven the formula for the sine of the sum of two real numbers 𝛼𝛼 and 𝛽𝛽 in the case where 0 < 𝛼𝛼 + 𝛽𝛽 < 𝜋𝜋

2. A proof for all real numbers 𝛼𝛼 and 𝛽𝛽 breaks down into cases that are proven similarly to the case

we have just seen. Although we are omitting the full proof, this formula holds for all real numbers 𝛼𝛼 and 𝛽𝛽.

4. Now, let’s prove our other conjecture, which is that the formula for the cosine of the sum of two numbers is

cos(𝛼𝛼 + 𝛽𝛽) = cos(𝛼𝛼) cos(𝛽𝛽) − sin(𝛼𝛼) sin(𝛽𝛽). Again, we will prove only the case when the two numbers are positive, and their sum is less than 𝜋𝜋

2. This

time, we will use the sine addition formula and identities from previous lessons instead of working through a geometric proof. Fill in the blanks in terms of 𝛼𝛼 and 𝛽𝛽: Let 𝛼𝛼 and 𝛽𝛽 be any real numbers. Then,

cos(𝛼𝛼 + 𝛽𝛽) = sin�𝜋𝜋2− (__________)�

= sin�(__________) − 𝛽𝛽�

= sin�(__________) + (−𝛽𝛽)�

= sin(__________) cos(−𝛽𝛽) + cos(__________) sin(−𝛽𝛽)

= cos(𝛼𝛼) cos(−𝛽𝛽) + sin(𝛼𝛼) sin(−𝛽𝛽)

= cos(𝛼𝛼) cos(𝛽𝛽)− sin(𝛼𝛼) sin(𝛽𝛽).

For any real numbers 𝛼𝛼 and 𝛽𝛽,

sin(𝛼𝛼 + 𝛽𝛽) = sin(𝛼𝛼)cos(𝛽𝛽) + cos(𝛼𝛼)sin(𝛽𝛽).

For all real numbers 𝛼𝛼 and 𝛽𝛽, cos(𝛼𝛼 + 𝛽𝛽) = cos(𝛼𝛼)cos(𝛽𝛽) − sin(𝛼𝛼)sin(𝛽𝛽).


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Formulas for 𝐬𝐬𝐬𝐬𝐬𝐬(𝜶𝜶 − 𝜷𝜷) and 𝐜𝐜𝐜𝐜𝐬𝐬(𝜶𝜶− 𝜷𝜷)

5. Rewrite the expression sin(𝛼𝛼 − 𝛽𝛽) as sin�𝛼𝛼 + (−𝛽𝛽)�. Use the rewritten form to find a formula for the sine of the difference of two angles, recalling that the sine is an odd function.

6. Now, use the same idea to find a formula for the cosine of the difference of two angles. Recall that the cosine is an even function.

For all real numbers 𝛼𝛼 and 𝛽𝛽, sin(𝛼𝛼 − 𝛽𝛽) = sin(𝛼𝛼)cos(𝛽𝛽) − cos(𝛼𝛼)sin(𝛽𝛽), and

cos(𝛼𝛼 − 𝛽𝛽) = cos(𝛼𝛼)cos(𝛽𝛽) + sin(𝛼𝛼)sin(𝛽𝛽).


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Other Identities

8. Derive a formula for tan(𝛼𝛼 + 𝛽𝛽) in terms of tan(𝛼𝛼) and tan(𝛽𝛽), where all of the expressions are defined. Hint: Use the addition formulas for sine and cosine.

9. Derive a formula for sin(2𝑢𝑢) in terms of sin(𝑢𝑢) and cos(𝑢𝑢) for all real numbers 𝑢𝑢.

10. Derive a formula for cos(2𝑢𝑢) in terms of sin(𝑢𝑢) and cos(𝑢𝑢) for all real numbers 𝑢𝑢.


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Homework Problem Set 1. Prove the formula

cos(𝛼𝛼 + 𝛽𝛽) = cos(𝛼𝛼) cos(𝛽𝛽) − sin(𝛼𝛼) sin(𝛽𝛽) for 0 < 𝛼𝛼 +𝛽𝛽 < 𝜋𝜋

2

using the rectangle 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 in the figure on the right and calculating 𝑀𝑀𝑀𝑀,𝑃𝑃𝑀𝑀, and 𝑃𝑃𝑀𝑀 in terms of 𝛼𝛼 and 𝛽𝛽.

2. Derive a formula for tan(2𝑢𝑢) for 𝑢𝑢 ≠ 𝜋𝜋4

+ 𝑘𝑘𝜋𝜋2

and 𝑢𝑢 ≠ 𝜋𝜋2



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3. Prove that cos(2𝑢𝑢) = 2cos2(𝑢𝑢) − 1 for any real number 𝑢𝑢.

4. Prove that 1cos(𝑥𝑥) − cos(𝑥𝑥) = sin(𝑥𝑥) ∙ tan(𝑥𝑥) for 𝑥𝑥 ≠ 𝜋𝜋

2+ 𝑘𝑘𝜋𝜋, for all integers 𝑘𝑘.


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5. Write as a single term: cos�𝜋𝜋4

+ 𝜃𝜃� + cos �𝜋𝜋4− 𝜃𝜃�.

6. Write as a single term: sin(25°) cos(10°) − cos(25°) sin(10°).

7. Write as a single term: cos(2𝑥𝑥) cos(𝑥𝑥) + sin(2𝑥𝑥) sin(𝑥𝑥).


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8. Write as a single term: sin(𝛼𝛼+𝛽𝛽)+sin(𝛼𝛼−𝛽𝛽)cos(𝛼𝛼) cos(𝛽𝛽) , where cos(𝛼𝛼) ≠ 0 and cos(𝛽𝛽) ≠ 0.

9. Prove that cos�3𝜋𝜋2

+ 𝜃𝜃� = sin(𝜃𝜃) for all values of 𝜃𝜃.

10. Prove that cos(𝜋𝜋 − 𝜃𝜃) = − cos(𝜃𝜃) for all values of 𝜃𝜃.


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HONORS ALGEBRA II/TRIGONOMETRY

Lesson 21: Ferris Wheels—Using Parametric Functions to Model Cyclical Behavior

Opening Exercise

1. Ernesto thinks that this is the graph of 𝑓𝑓(𝑥𝑥) = 4 sin �𝑥𝑥 − 𝜋𝜋2�. Danielle thinks it is the graph of 𝑓𝑓(𝑥𝑥) = 4cos(𝑥𝑥).

Who is correct and why?

Lesson 21: Ferris Wheels—Using Parametric Functions to Model Cyclical Behavior Unit 7: Transformations & Identities

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Exploratory Challenge/Exercises 1–5

A carnival has a Ferris wheel that is 50 feet in diameter with 12 passenger cars. When viewed from the side where passengers board, the Ferris wheel rotates counterclockwise and makes two full turns each minute. Riders board the Ferris wheel from a platform that is 15 feet above the ground. We will use what we have learned about periodic functions to model the position of the passenger cars from different mathematical perspectives. We will use the points on the circle in the diagram on the right to represent the position of the cars on the wheel.

2. For this exercise, we will consider the height of a passenger car to be the vertical displacement from the horizontal line through the center of the wheel and the co-height of a passenger car to be the horizontal displacement from the vertical line through the center of the wheel. a. Let 𝜃𝜃 = 0 represent the position of car 1 in the diagram above. Sketch the graphs of the co-height

and the height of car 1 as functions of 𝜃𝜃, the number of radians through which the car has rotated.

b. What is the amplitude, |𝐴𝐴|, of the height and co-height functions for this Ferris wheel? c. Let 𝑋𝑋(𝜃𝜃) represent the co-height function after rotation by 𝜃𝜃 radians, and let 𝑌𝑌(𝜃𝜃) represent the

height function after rotation by 𝜃𝜃 radians. Write functions 𝑋𝑋 for the co-height and 𝑌𝑌 for the height in terms of 𝜃𝜃.


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d. Graph the functions 𝑋𝑋 and 𝑌𝑌 from part (c) on a graphing calculator set to parametric mode. Use a viewing window [−48, 48] × [−30, 30]. Sketch the graph below.

e. Why did we choose the symbols 𝑋𝑋 and 𝑌𝑌 to represent the co-height and height functions?

f. Evaluate 𝑋𝑋(0) and 𝑌𝑌(0), and explain their meaning in the context of the Ferris wheel.

g. Evaluate 𝑋𝑋 �𝜋𝜋2� and 𝑌𝑌 �𝜋𝜋

2�, and explain their meaning in the context of the Ferris wheel.


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3. The model we created in Exercise 2 measures the height of car 1 above the horizontal line through the center of the wheel. We now want to alter this model so that it measures the height of car 1 above the ground. a. If we measure the height of car 1 above the ground

instead of above the horizontal line through the center of the wheel, how will the functions 𝑋𝑋 and 𝑌𝑌 need to change?

b. Let 𝜃𝜃 = 0 represent the position of car 1 in the diagram to the right. Sketch the graphs of the co-

height and the height of car 1 as functions of the number of radians through which the car has rotated, 𝜃𝜃.

c. How are the graphs from part (b) related to the graphs from Exercise 2(a)?


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d. From this perspective, find the equations for the functions 𝑋𝑋 and 𝑌𝑌 that model the position of car 1

with respect to the number of radians the car has rotated.

e. Change the viewing window on your calculator to [−60, 60] × [−5, 70], and graph the functions 𝑋𝑋 and 𝑌𝑌 together. Sketch the graph.

f. Evaluate 𝑋𝑋(0) and 𝑌𝑌(0), and explain their meaning in the context of the Ferris wheel.

g. Evaluate 𝑋𝑋 �𝜋𝜋2� and 𝑌𝑌 �𝜋𝜋



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4. In reality, no one boards a Ferris wheel halfway up; passengers board at the bottom of the wheel. To truly model the motion of a Ferris wheel, we need to start with passengers on the bottom of the wheel. Refer to the diagram below.

a. Let 𝜃𝜃 = 0 represent the position of car 1 at the bottom of the wheel in the diagram above. Sketch the graphs of the height and the co-height of car 1 as functions of 𝜃𝜃, the number of radians through which the car has rotated from the position at the bottom of the wheel.

b. How are the graphs from part (a) related to the graphs from Exercise 3(b)?


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c. From this perspective, find the equations for the functions 𝑋𝑋 and 𝑌𝑌 that model the position of car 1

with respect to the number of radians the car has rotated.

d. Graph the functions 𝑋𝑋 and 𝑌𝑌 from part (c) together on the graphing calculator. Sketch the graph. How is this graph different from the one from Exercise 3(e)?

e. Evaluate 𝑋𝑋(0) and 𝑌𝑌(0), and explain their meaning in the context of the Ferris wheel.

f. Evaluate 𝑋𝑋 �𝜋𝜋2� and 𝑌𝑌 �𝜋𝜋



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5. Finally, it is not very useful to track the position of a Ferris wheel as a function of how much it has rotated. It would make more sense to keep track of the Ferris wheel as a function of time. Recall that the Ferris wheel completes two full turns per minute.

a. Let 𝜃𝜃 = 0 represent the position of car 1 at the bottom of the wheel. Sketch the graphs of the co-height and the height of car 1 as functions of time.


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b. The co-height and height functions from part (a) can be written in the form 𝑋𝑋(𝑡𝑡) = 𝐴𝐴 cos�𝜔𝜔(𝑡𝑡 − ℎ)� + 𝑘𝑘 and 𝑌𝑌(𝑡𝑡) = 𝐴𝐴 sin�𝜔𝜔(𝑡𝑡 − ℎ)� + 𝑘𝑘. From the graphs in part (a), identify the values of 𝐴𝐴, 𝜔𝜔, ℎ, and 𝑘𝑘 for each function 𝑋𝑋 and 𝑌𝑌.

c. Write the equations 𝑋𝑋(𝑡𝑡) and 𝑌𝑌(𝑡𝑡) using the values you identified in part (b).

d. In function mode, graph your functions from part (c) on a graphing calculator for 0 < 𝑡𝑡 < 2, and compare against your sketches in part (a) to confirm your equations.

e. Explain the meaning of the parameters in your equation for 𝑋𝑋 in terms of the Ferris wheel scenario.

f. Explain the meaning of the parameters in your equation for 𝑌𝑌 in terms of the Ferris wheel scenario.


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6. In parametric mode, graph the functions 𝑋𝑋 and 𝑌𝑌 from Exercise 4(c) on a graphing calculator for 0 ≤ 𝑡𝑡 ≤ 12.

a. Sketch the graph. How is this graph different from the graph in Exercise 4(d)?

b. What would the graph look like if you graphed the functions 𝑋𝑋 and 𝑌𝑌 from Exercise 4(c) for 0 ≤ 𝑡𝑡 ≤ 14?

Why?


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c. Evaluate 𝑋𝑋(0) and 𝑌𝑌(0), and explain their meaning in the context of the Ferris wheel.

d. Evaluate 𝑋𝑋 �18� and 𝑌𝑌 �1


Practice Exercise 7–10

7. You are in car 1, and you see your best friend in car 3. How much higher than your friend are you when you reach the top?


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8. Find an equation of the function 𝐻𝐻 that represents the difference in height between you in car 1 and your friend in car 3 as the wheel rotates through 𝜃𝜃 radians, beginning with 𝜃𝜃 = 0 at the bottom of the wheel.

9. Find an equation of the function that represents the difference in height between car 1 and car 3 with respect to time, 𝑡𝑡, in minutes. Let 𝑡𝑡 = 0 minutes correspond to a time when car 1 is located at the bottom of the Ferris wheel. Assume the wheel is moving at a constant speed starting at 𝑡𝑡 = 0.

10. Use a calculator to graph the function 𝐻𝐻 in Exercise 9 for 0 ≤ 𝑡𝑡 ≤ 2. What type of function does this appear to be? Does that make sense?


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Homework Problem Set 1. In the classic novel Don Quixote, the title character famously

battles a windmill. In this problem, you will model what happens when Don Quixote battles a windmill, and the windmill wins. Suppose the center of the windmill is 20 feet off the ground, and the sails are 15 feet long. Don Quixote is caught on a tip of one of the sails. The sails are turning at a rate of one counterclockwise rotation every 60 seconds. a. Explain why a sinusoidal function could be used to model

Don Quixote’s height above the ground as a function of time.

b. Sketch a graph of Don Quixote’s height above the ground as a function of time. Assume 𝑡𝑡 = 0 corresponds to a time when he was closest to the ground. What are the amplitude, period, and midline of the graph?

c. Model Don Quixote’s height 𝐻𝐻 above the ground as a function of time 𝑡𝑡 since he was closest to the ground.


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d. After 1 minute and 40 seconds, Don Quixote fell off the sail and straight down to the ground. How far did he fall?

2. The High Roller, a Ferris wheel in Las Vegas, Nevada, opened in March 2014. The 550 ft. tall wheel has a diameter of 520 feet. A ride on one of its 28 passenger cars lasts 30 minutes, the time it takes the wheel to complete one full rotation. Riders board the passenger cars at the bottom of the wheel. Assume that once the wheel is in motion, it maintains a constant speed for the 30-minute ride and is rotating in a counterclockwise direction. a. Sketch a graph of the height of a passenger car on the High Roller as a function of the time the ride

began.

b. Write a sinusoidal function 𝐻𝐻 that represents the height of a passenger car 𝑡𝑡 minutes after the ride begins.


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c. Explain how the parameters of your sinusoidal function relate to the situation.

d. If you were on this ride, how high would you be above the ground after 20 minutes?

e. Suppose the ride costs $25. How much does 1 minute of riding time cost? How much does 1 foot of riding distance cost? How much does 1 foot of height above the ground cost?

f. What are some of the limitations of this model based on the assumptions that we made?


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3. Once in motion, a pendulum’s distance varies sinusoidally from 12 feet to 2 feet away from a wall every 12 seconds. a. Sketch the pendulum’s distance 𝐷𝐷 from the wall over a 1-minute interval as a function of time 𝑡𝑡.

Assume 𝑡𝑡 = 0 corresponds to a time when the pendulum was furthest from the wall.

b. Write a sinusoidal function for 𝐷𝐷, the pendulum’s distance from the wall, as a function of the time since it was furthest from the wall.

c. Identify two different times when the pendulum was 10 feet away from the wall. (Hint: Write an equation, and solve it graphically.)


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4. The height in meters relative to the starting platform height of a car on a portion of a roller coaster track

is modeled by the function 𝐻𝐻(𝑡𝑡) = 3 sin �𝜋𝜋4

(𝑡𝑡 − 10)� − 7. It takes a car 24 seconds to travel on this

portion of the track, which starts 10 seconds into the ride. a. Graph the height relative to the starting platform as a function of time over this time interval.

b. Explain the meaning of each parameter in the height function in terms of the situation.

5. Given the following function, use the parameters to formulate a real-world situation involving one

dimension of circular motion that could be modeled using this function. Explain how each parameter of the function relates to your situation.

𝑓𝑓(𝑥𝑥) = 10 sin�𝜋𝜋8

(𝑥𝑥 − 3)�+ 15


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