unit 9: trigonometry notes packet -...

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Math 2 Name _______________________________

Unit 9: Trigonometry Notes Packet

9-1: Intro to Right Triangle Trigonometry

9-2: Right Triangle Trigonometry

9-3: Right Triangle Trig Applications

9-4: Special Right Triangles (The Unit Circle)

9-5: The Unit Circle (Degree Measure)

9-6: The Unit Circle (Radian Measure)

9-7: Law of Sines and Law of Cosines

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9-1: Introduction to Right Triangle Trigonometry https://www.youtube.com/watch?v=t2uPYYLH4Zo Right Triangle Basics Q: What is the definition of a “right triangle” ? ____________________________________

There are six parts to a right triangle. A right triangle has two _______________ angles and one _______________ angle. The longest side of a right triangle called the _______________ and the two shorter sides are called _______________ .

If the measures of three parts of a right triangle are known, then the remaining three parts can be found using trigonometry and/or the Pythagorean Theorem.

A side c side b C B side a

The three trig functions that can used to the find missing values on a right triangle are …

sine (sin) , cosine (cos) , tangent (tan) … and they are found on your calculator.

Q: Do you remember SOHCAHTOA ? What do all the letters stand for?

S ____________ C ____________ T ____________ O ____________ A ____________ H ____________

sin = cos = tan =

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Example: Find the ratio (fraction) for each, using the triangle below. A c b B a C sin A = ______ cos A = ______ tan A = ______ sin B = ______ cos B = ______ tan B = ______

Q: Notice any patterns? __________________________________ ____________________________________________________

Example: Find the ratio (fraction) for each, using the triangle below. Uncle Pythagoras may help. A 5 3 B C sin A = ______ cos A = ______ tan A = ______ sin B = ______ cos B = ______ tan B = ______ Example: Find the ratio (fraction) for each, using the triangle below. Uncle Pythagoras may help. B 5 A 12 C sin A = ______ cos A = ______ tan A = ______ sin B = ______ cos B = ______ tan B = ______

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Not every right triangle can give you a “Pythagorean Triple.” For example, the sides are 6 cm, 8 cm, and 10 cm or the sides are 5 cm, 12 cm , and 13 cm. You may have to simplify some radicals. Here’s some ( REVIEW ) practice …

Completely simplify each radical expression. 1.) 24 2.) 200 3.) 60 4.) 148 5.) 32 6.) 280 7.) 232 8.) 224 9.) 253 10.) 266

11.) 2

3 12.) 3

5 13.) 6

12 14.) 6

14

Example: Find the “simplified” ratio (fraction) for each, using the triangle below. Uncle Pythagoras may help. A 6 B 14 C sin A = ______ cos A = ______ tan A = ______

sin B = ______ cos B = ______ tan B = ______

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Lesson 9-1 Intro to Right Triangle Trigonometry Homework 1.) Write the ratio (fraction) for each trig function of each triangle. A X a Y p t b c C s B Z sin A = ________ sin B = ________ sin X = ________ sin Z = ________ cos A = ________ cos B = ________ cos X = ________ cos Z = ________ tan A = ________ tan B = ________ tan X = ________ tan Z = ________ 2.) Find the exact (simplified) value for each of the six trig functions for and .

__________sin 2

__________cos

5 __________tan

__________sin __________cos __________tan 3.) Find the exact (simplified) value for each of the six trig functions for and .

14 __________sin

__________cos

8 __________tan

__________sin __________cos __________tan

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4.) Find the exact (simplified) value for each of the six trig functions for .

6

15

__________sin __________cos __________tan

__________cot __________sec __________csc 5.) If 11

7cos , find the exact, simplified ratio ( for ) for each of the remaining trig functions.

__________sin __________tan

6.) If 7

24tan , find the exact, simplified ratio ( for ) for each of the remaining trig functions.

__________sin

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9-2: Right Triangle Trigonometry

Finding Missing Sides & Angles Using “SOHCAHTOA” Example: Set up two different equations to find the length of side BC. The set up depends on where (what

“acute” angle) you are standing. Then find BC. A

68 18 in

C B

1.) ________________________ 2.) ________________________ Practice Examples: Solve each triangle (find all the missing parts). You need to draw the triangle in each

scenario to assist you. 1.) Given right ABC with 90Cm . Solve the triangle if 51Am and AC = 9.

2.) Given right DEF with 90Dm . Solve the triangle if 32Em and EF = 11. 3.) Given right QRS with 90Sm . Solve the triangle if 42Qm and RS = 19.

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Q: How do you find the measure of a missing angle when given only two sides? Example: Find Bm . A

22 in

C 17 in B Set-up: ________________________ Example: Find Am . A

4 in

C 12 in B Set-up: ________________________ Practice Examples: Solve each triangle (find all the missing parts). You need to draw the triangle in each

scenario to assist you. 1.) Given right ABC with 90Cm . Solve the triangle if AB = 15 and AC = 11. ______________Bm ______________Am BC = ________________ 2.) Given right DEF with 90Dm . Solve the triangle if DE = 13 and DF = 18. ______________Em ______________Fm EF = ________________

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3.) Given right QRS with 90Sm . Solve the triangle if QR = 23 and RS = 12. ______________Qm ______________Rm QS = ________________

Lesson 9-2 Right Triangle Trigonometry Homework In #1-4, find the side length indicated. Round to the nearest tenth. For #5 – 8, find the measure of each angle or side indicated. Round to the nearest tenths place. 5) 6)

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7) 8) 9) Solve the triangle. (Decimal answers are appropriate.)

A

29 C B 16 cm 10) Solve the triangle. (Decimal answers are appropriate.)

A 23 cm 9 cm

C B Challenge: Draw what you read and set up a sine, cosine, or tangent equation to find the height of the building. 11.) You have decided to base jump off of the tallest building in North America. When standing at the top of

it you can see an ambulance on the street below at an angle of depression of 78 . If the ambulance is parked 480 ft from the base of the building, how tall is the building? (Your line of sight is 5 ft above your feet.)

Equation (set-up): ____________________________ Ans: ___________________

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9-3: Right Triangle Trig Applications Angle of Elevation line of sight horizontal

Angle of Depression horizontal line of sight Draw each scenario and find the missing measurement. Example 1: You lean a ladder 6.7 meters long against the wall. It makes an angle of 63° with the level

ground. How high above the ground is the top of the ladder? Example 2: Suppose you have been assigned to measure the height of the local water tower. Climbing makes

you dizzy, so you decide to do the whole job at ground level. From a point 47.3 ft from the base of the water tower, you find that you must look up at an angle of 53° to see the top of the tower. How tall is the tower?

(Your line of sight is 5 ft above your feet.)

The Angle of Elevation is the angle from the horizontal to your line of sight. (i.e. you are looking upwards at the object)

The Angle of Depression is the angle from the horizontal to the line of sight. (i.e. you are looking downwards at the object)

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Example 3: The tallest freestanding structure in the world is the 553 meter tall CN tower in Toronto, Ontario.

Suppose that at a certain time of day it casts a shadow 1100 meters long on the ground. What is the angle of elevation of the sun at that time of day?

Lesson 9-3 Right Triangle Trig Applications Homework 1.) You are standing 55 ft from the base of a large building. As you look up at the building you see a small

stone statue at an angle of elevation of 78 . How tall is the building if the statue has been placed exactly in the middle of the building?

(line of sight is 5 ft above your feet) 2.) A 312-ft-tall radio tower stands alone in field off of the highway. The tower is mainly supported by

large cables that are connected to the ground and also to a stabilization brace that is located three-fourths of the way up the tower. How long are the cables if they make a 65 - angle with the ground?

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3.) A 75 tall flagpole breaks at a certain point, but remains attached. The top of the upper part of the pole hits the ground 48 ft from the base of the flagpole. How tall is the flagpole if the angle created at the breaking point is 70 ?

4.) A helicopter is hovering above Nationwide Arena such the pilot’s line of sight is at an altitude of 850 ft.

He spots a wreck on 315 North exactly two miles from the arena. What is the pilot’s angle of depression when he is looking at the wreck?

5.) At 10:37 a.m., the angle of elevation of the sun is 73 . At this particular time, what is the length of the

shadow cast by a 480 ft-tall building?

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9-4: Special Right Triangles

Do you see a pattern? 45-45-90 Triangles A C B Practice: Find the missing side lengths. 1) 2) 3)

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Do you see a pattern? 30-60-90 Triangles Practice: Find the missing side lengths. 4) 5) 6)

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Special Right Triangle Properties:

7) 8) 9) 10)

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Lesson 9-4 Special Right Triangles Homework Find the missing side lengths. Leave your answers as radicals in simplified form.

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9-5 Part 1: Special Right Triangles and The Unit Circle We can use these “special” angles to build the “unit” circle. The ____________ of any “unit” circle is the _______________ of either one of the two special right triangles. The radius of a “unit” circle has a length of ____ unit Fill in all of the special angle measures (in degrees) around the unit circle.

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9-5: The Unit Circle (Degree Measure) Special ordered pairs are strategically placed around the unit circle at these special angles. The x-coordinate of each of those ordered pairs in the cos of that angle ( cos ) and the y-coordinate of each of those ordered pairs is the sin of that angle ( sin ). The tangent of each of those angles can be found by dividing the sine of the angle by the cosine of the angle …

cossintan

Find the coordinates (exact values) for all of the ordered pairs along the unit circle below.

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The Unit Circle, learn it and understand it. Complete in degrees only.

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Unit Circle Example Problems: Find the exact value of each. 1.) 315sin = ________ 2.) 150tan = ________ 3.) 240cos = ________ 4.) 45cos = ________ 5.) 120sin = ________ 6.) 270tan = ________ 7.) 180sin = ________ 8.) 150cos = ________ 9.) 300tan = ________ We can also work with angles that measure more than 360 and angles that have negative degree-measures. 10.) )315sin( = ________ 11.) )150tan( = ________ 12.) 390cos = ________ 13.) 450cos = ________ 14.) 750sin = ________ 15.) )630tan( = ________ 16.) )180sin( = ________ 17.) 330cos = ________ 18.) 300tan = ________

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8. Find θ. __________________ 9. Find y __________________ 10. Find α. __________________ 11. Find x. __________________

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9-6: The Unit Circle (Radian Measure) A radian is another unit of measurement for an angle. Here’s a visual definition of a radian. Can you define it in words? = 1 radian 9 cm Degree-Radian Conversion … Q: What is the (exact) answer for the circumference of a “unit” circle? ___________ … thus __________ = __________ and __________ = __________ radians degrees radians degrees This is our conversion factor. Example: Convert each to degrees.

1.) 32 2.) 4

7 3.) 95 4.) 6

11 Example: Convert each to radians (in terms of ). 1.) 300 2.) 150 3.) 100 4.) 270 Introduction to the “Three Pizzas” …

I.) 6 - slice II.) 4

- slice III.) 3 - slice

9 cm

Definition:

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Teaching Examples:

1.) cos 35 = __________ 2.) tan 6

7 = __________ 3.) sin 43 = __________

4.) tan 29 = __________ 5.) sin 6

23 = __________ Examples: Find the exact value for each.

1.) tan 43 = __________ 2.) cos 2

3 = __________ 3.) sin 65 = __________

4.) cos 613 = __________ 5.) sin 4

5 = __________ 6.) tan 37 = __________

Complete in Radians and Degrees.

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9-7: Law of Sines and Law of Cosines Triangle Extension Intro Example: Find the length of side QR to the nearest tenth of an inch. R Q 35 12 in 130

S

Law of ________________ FORMULA: Law of ________________ FORMULA: Q: Which of the following triangle scenarios work with the Law of Sines? Q: Which of the following triangle scenarios work with the Law of Sines? SAS ASA SSS AAS Example Problems: 1.) DEF with d = 24 , e = 36 , and f = 14, find the missing parts.

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2.) ABC with b = 11.3 , c = 5.8 , and 52A , find the missing parts. 3.) RST with s = 15 , t = 19 , and 61T , find the missing parts. 4.) Find the height of the mountain.

unit 9: trigonometry notes packet -...

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