homework, page 366

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 1

Homework, Page 366

Find the values of all six trigonometric functions of the angle x.

1.5 4

3

x

4 3 4sin ;cos ; tan

5 5 35 5 3

csc ;sec ;cot4 3 4

x x x

x x x


Homework, Page 366Find the values of all six trigonometric functions of the angle x.

5.

2 2 2 27 11 150 5 6

7 7 6 11 11 6 7sin ;cos ; tan

30 30 115 6 5 6

5 6 5 6 11csc ;sec ;cot

7 11 7

c a b

x x x

x x x

7

11

x


Homework, 66Assume that θ is an acute angle in a right triangle satisfying the given condition. Evaluate the remaining trigonometric functions.

9. 3sin

7

2 23sin 49 9 2 10

7

2 10 3 3 10cos ; tan

7 202 10

7 7 7 10 2 10csc ;sec ;cot

3 20 32 10

a c b

x


Homework, Page 366Assume that θ is an acute angle in a right triangle satisfying the given condition. Evaluate the remaining trigonometric functions.

13. 5tan

9

2 25tan 25 81 106

9

5 5 106 9 3 106sin ;cos

106 106106 106

106 106 9csc ;sec ;cot

5 9 5

c a b

x


Homework, Page 366Assume that θ is an acute angle in a right triangle satisfying the given condition. Evaluate the remaining trigonometric functions.

17. 23csc

9

2 223csc 529 81 448 8 7

9

9 8 7 9 9 7sin ;cos ; tan

23 23 568 7

23 23 7 8 7sec ;cot

56 98 7

a c b


Homework, Page 366Evaluate without using a calculator.

21. cot6

3cos6 2cot 3

16sin

26


Homework, Page 366Evaluate using a calculator. Given an exact value.

25. sec45

1 2 2 2sec45 2

22 2

2


Homework, Page 366Evaluate using a calculator, giving answers to three decimal places.

29. sin 74sin 74 0.961



33. tan12

tan 0.26812



37. cot 0.89cot 0.89 0.810


Homework, Page 366Without a calculator, find the acute angle θ that satisfies the equation. Give θ in both degrees and radians.

41. 1sin

2

1sin 30

2 6


Homework, Page 366Without a calculator, find the acute angle θ that satisfies the equation. Give θ in both degrees and radians.

45. sec 2

1sec 2 cos 60

2 3


Homework, Page 366Solve for the variable shown.

49.

15 15sin34 26.824

sin34x

x

x15

34



53.

6 6sin35 10.461

sin35y

y

6

35

y



57.55 15.58a

x

y

a

b

c

55 15.58 cos 27.163cos

35 tan tan 22.251

a aa c

c

bb a

a


Homework, Page 36661. A guy wire from the top of a radio tower forms a 75º angle with the ground at a 55 ft distance from the foot of the tower. How tall is the tower?

75 55 tan

tan 55 tan 75 205.263 ft

ba

a

b a

55

75


Homework, Page 36665. A surveyor wanted to measure the length of a lake. Two assistants, A and C, positioned themselves at opposite ends of the lake and the surveyor positioned himself 100 feet perpendicular to the line between the assistants and on the perpendicular line from the assistant C. If the angle between his lines of sight to the two assistants is 79º12‘42“, what is the length of the lake?

75 12 42 tan100

42 12.775 12 42 75 12 75 75.212

60 60

100 tan 75 12 42 378.797 ft

AC AC

BC

AC

75 12' 42"

100

ft

A C

B


Homework, Page 36669. Which of the following expressions does not

represent a real number?

a. sin 30º

b. tan 45º

c. cod 90º

d. csc 90º

e. sec 90º

1

1

sin30 0.5; tan 45 1

cos90 0;csc90 1

csc90 0


Homework, Page 36673. The table is a simplified trig table. Which column is the values for the sine, the cosine, and the tangent functions?

The second column is tangent values, because tangent can be greater than one, the third is sine values, because they are increasing and the fourth column is cosine values because they are decreasing.

Angle ? ? ?

40º 0.8391 0.6428 0.7660

42 º 0.9004 0.6691 0.7431

44 º 0.9657 0.6047 0.7191

46 º 1.0355 0.7191 0.6047

48 º 1.1106 0.7431 0.6691

50 º 1.1917 0.7660 0.6428

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

4.3

Trigonometry Extended: The Circular Functions


What you’ll learn about

Trigonometric Functions of Any Angle Trigonometric Functions of Real Numbers Periodic Functions The 16-point unit circle

… and whyExtending trigonometric functions beyond triangle ratios opens up a new world of applications.


Leading Questions

We may substitute any real number n for θ in any trig function and find the value of the function.Cosine is negative in the fourth quadrant.Coterminal angles have the same measure.Quadrantal angles have their terminal sides in the center of the quadrants.The period of a trig function tells us how often it takes on identical values.

Slide 4- 22


Initial Side, Terminal Side


Positive Angle, Negative Angle


Coterminal Angles

Two angles in an extended angle-measurement system can have the same initial side and the same terminal side, yet have different measures. Such angles are called coterminal angles.


Example Finding Coterminal Angles

Find a positive angle and a negative angle that are coterminal

with 45 .


Example Finding Coterminal Angles

Find a positive angle and a negative angle that are coterminal

with .6


Example Evaluating Trig Functions Determined by a Point in Quadrant I

Let be the acute angle in standard position whose terminal

side contains the point (3,5). Find the six trigonometric functions

of .


Trigonometric Functions of any Angle

2 2

Let be any angle in standard position and let ( , )

be any point on the terminal side of the angle (except

the origin). Let denote the distance from ( , ) to

the origin, i.e., let . Then

P x y

r P x y

r x y

sin csc ( 0)

cos sec ( 0)

tan ( 0) cot ( 0)

y ry

r y

x rx

r xy x

x yx y


Evaluating Trig Functions of a Nonquadrantal Angle θ

1. Draw the angle θ in standard position, being careful to place the terminal side in the correct quadrant.

2. Without declaring a scale on either axis, label a point P (other than the origin) on the terminal side of θ.

3. Draw a perpendicular segment from P to the x-axis, determining the reference triangle. If this triangle is one of the triangles whose ratios you know, label the sides accordingly. If it is not, then you will need to use your calculator.

4. Use the sides of the triangle to determine the coordinates of point P, making them positive or negative according to the signs of x and y in that particular quadrant.

5. Use the coordinates of point P and the definitions to determine the six trig functions.


Signs of Trigonometric Functions

Slide 4- 31


Reference Angles

The acute angle made by the terminal side of an angle and the x-axis is called the reference angle.

The absolute value of each trig function is equal to the absolute value of the same trig function of the reference angle in the first quadrant. The sign of the trig function is determined by the quadrant in which the terminal side lies.

Slide 4- 32


Example Evaluating More Trig Functions

Find sin 210 without a calculator.


Example Using one Trig Ratio to Find the Others

Find sin and cos , given tan 4 / 3 and cos 0.


Unit Circle

The unit circle is a circle of radius 1 centered at the origin.


Trigonometric Functions of Real NumbersLet be any real number, and let ( , ) be the point

corresponding to when the number line is wrapped

onto the unit circle as described above. Then

1sin csc ( 0)

cos

t P x y

t

t y t yy

t x

1

sec ( 0)

tan ( 0) cot ( 0)

t xx

y xt x t yx y


Periodic Function

A function ( ) is if there is a positive number such that

( ) ( ) for all values of in the domain of . The smallest such

number is called the of the function.

y f t c

f t c f t t f

c

periodic

period


The 16-Point Unit Circle


Following Questions

Graphs of the sine function may be stretched vertically, but not horizontally.

Horizontal stretches of the cosine function are the result of changes in its period.

Horizontal translations of the sine function are the result of phase shifts.

Sinusoids are functions whose graphs have the shape of the sine curve.

Sinusoids may be used to model periodic behavior.

Slide 4- 39


Homework

Homework Assignment #28 Review Section 4.3 Page 381, Exercises: 1 – 69 (EOO) Quiz next time


4.4

Graphs of Sine and Cosine: Sinusoids


Quick Review

1

State the sign (positive or negative) of the function in each quadrant.

1. sin

2. cot

Give the radian measure of the angle.

3. 150

4. 135

5. Find a transformation that will transform the graph of

x

x

y

2

to

the graph of 2 .

x

y x


Quick Review Solutions

+,+, ,

+, ,+

State the sign (positive or negative) of the function in each quadrant.

1. sin

2. cot

Give the radian measure of the angle.

3. 150

,

5 /6

3 /4

4. 135

5. Find a transformation th

x

x

1

2

at will transform the graph of to

the vertgr icaph allyof 2 stretch by 2.

y x

y x


What you’ll learn about

The Basic Waves Revisited Sinusoids and Transformations Modeling Periodic Behavior with Sinusoids

… and why

Sine and cosine gain added significance when

used to model waves and periodic behavior.


Sinusoid

A function is a if it can be written in the form

( ) sin( ) where , , , and are constants

and neither nor is 0.

f x a bx c d a b c d

a b

sinusoid


Amplitude of a Sinusoid

The of the sinusoid ( ) sin( ) is | |.

Similarly, the amplitude of ( ) cos( ) is | |.

Graphically, the amplitude is half the distance between the

trough and the crest of the wave

f x a bx c d a

f x a bx c d a

amplitude

.


Example Finding Amplitude

Find the amplitude of each function and use the language of transformations to describe how the graphs are related.

(a) (b) (c) 1 siny x 2 2siny x 1

1sin

3y x


Period of a Sinusoid

The of the sinusoid ( ) sin( ) is

2 / | | . Similarly, the period of ( ) cos( )

is 2 / | | . Graphically, the period is the length of one

full cycle of the wave.

p f x a bx c d

p b f x a bx c d

p b

period


Example Finding Period and Frequency

Find the period and frequency of each function and use the language of transformations to describe how the graphs are related.

(a) (b) (c) 1 siny x 2 2sin 2y x 1 3sin3

xy


Example Horizontal Stretch or Shrink and Period

Find the period of sin and use the language of 2

transformations to describe how the graph relates to

sin .

xy

y x


Frequency of a Sinusoid

The of the sinusoid ( ) sin( )

is | | / 2 1 . Similarly, the frequency of

( ) cos( ) is | | / 2 1 . Graphically,

the frequency is the number of complete cycles the wave

c

f f x a bx c d

f b p

f x a bx c d f b p

frequency

ompletes in a unit interval.


Example Combining a Phase Shift with a Period Change

Construct a sinusoid with period /3 and amplitude 4

that goes through (2,0).


Graphs of Sinusoids The graphs of sin( ( )) and cos( ( ))

(where 0 and 0) have the following characteristics:

amplitude = | | ;

2period = ;

| |

| |frequency = .

2When complared to the graphs of sin and

y a b x h k y a b x h k

a b

a

b

b

y a bx

cos ,

respectively, they also have the following characteristics:

a phase shift of ;

a vertical translation of .

y a bx

h

k


Constructing a Sinusoidal Model using Time

1. Determine the maximum value and minimum value .

The amplitude of the sinusoid will be , and 2

the vertical shift will be .2

2. Determine the period , the time interval of a single cy

M m

M mA A

M mC

p

cle

of the periodic function. The horizontal shrink (or stretch)

2will be .B

p


Constructing a Sinusoidal Model using Time

3. Choose an appropriate sinusoid based on behavior

at some given time . For example, at time :

( ) cos( ( )) attains a maximum value;

( ) cos( ( )) attains a minimum value;

( ) sin( (

T T

f t A B t T C

f t A B t T C

f t A B t

)) is halfway between a minimum

and a maximum value;

( ) sin( ( )) is halfway between a maximum

and a minimum value.

T C

f t A B t T C


Example Constructing a Sinusoidal Model

On a certain day, high tide occurs at 7:12 AM and the

water depth is measured at 15 ft. On the same day, low

tide occurs at 1:24 and the water depth measures 8 ft.

(a) Write a sinusoidal function modeling the tide.

(b) What is the approximate depth of water at 11:00 AM?

At 3:00 PM?


Example Constructing a Sinusoidal Model

(c) At what time did the first low tide occur? The second

high tide?

homework, page 366

Documents