Homework, Page 381

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

Homework, Page 381Identify the one angle that is not coterminal with the others.

1. 150 , 510 , 210 , 450 , 870

150

510 360 150

210 360 150

450 360 90 150

870 360 510 360 150

450 is not coterminal with the others


Homework, Page 381Evaluate the six trigonometric functions of the angle θ.

5.

2 22 2 1 1 2

1 2 2sin csc 2

2 12

1 2 2cos sec 2

2 121 1

tan 1 cot 11 1

r a b

x

y

(-1, -1)


Homework, 81Point P is on the terminal side of angle θ. Evaluate the six trigonometric functions for θ. If the function is undefined, write undefined.

9. 0,5P

902

sin 1 csc 1

1cos 0 sec undefined

01 0

tan undefined cot 00 1


Homework, Page 381State the sign (+ or –) of (a) sin t, (b) cos t (c) tan t for values of t in the interval given.

13. 0,2

sin

cos

tan

a t

b t

c t


Homework, Page 381Determine the sign (+ or –) of the given value without a calculator.

17. cos143cos143

90 143 180, Quadrant II, cosine negative


Homework, Page 381Choose the point on the terminal side of θ.

21.

(a) (2, 2) (b) (c)

Choice (a) as tan 45º = 1.

45

1, 3 3,1


Homework, Page 381Evaluate without using a calculator by using ratios in a reference triangle.

25. cos120120 180 60 cos is negative in Quadrant II

1cos120 cos60

2



29. 13

sin6

13 12 1 1sin sin sin

6 6 6 6 2



33. 23

cos6

23 24cos cos cos

6 6 6 6

Cosine is positive in Quadrant IV

3cos

6 2


Homework, Page 381Find (a) sin θ, (b) cos θ, and (c) tan θ for the given quadrantal angle. If the value is undefined, write undefined.

37. 450

450 360 90 360 270

sin 270 1

cos270 0

tan 270 undefined

a

b

c


Homework, Page 381Find (a) sin θ, (b) cos θ, and (c) tan θ for the given quadrantal angle. If the value is undefined, write undefined.

41. 7

2

7 4 3 4

2 2 2 2 2

sin 12

cos 02

tan undefined2

a

b

c


Homework, Page 381Evaluate without using a calculator.

45. 2

Find tan and sec if sin and cos 05

2 25 2 21

1 1 5 5 21sec

cos 2121 21

5

2 2 21tan

2121

x


Homework, Page 381Evaluate by using the period of the function.

49. sin 49,0006

49000sin 49,000 24500

6 2

1sin 49,000 sin

6 6 2


Homework, Page 381

53. Use your calculator to evaluate the expressions in Exercises 49 – 52. Does your calculator give the correct answer. Many miss all four. Give a brief explanation why.

The calculator algorithms apparently do recognize large multiples of pi and end up evaluating at nearby values.


Homework, Page 381

57. A weight suspended from a spring is set into motion. Its displacement d from equilibrium is modeled by the equation where d is the displacement in inches and t is the time in seconds. Find the displacement at the given time.

(a) t = 0

(b) t = 3

0.20.4 cos4td e t

0.2 00.4 cos4 0 0.4 1 1 0.4d e

0.2 30.4 cos4 3 0.185d e in


Homework, Page 381

61. If θ is an angle in a triangle such that cos θ < 0, then θ is an obtuse angle. Justify your answer.

True. An obtuse angle in the standard position would have its terminal side in the second quadrant and cosine is negative in the second quadrant.


Homework, Page 381

65. The range of the function

a. [1]

b. [-1, 1]

c. [0, 1]

d. [0, 2]

e. [0, ∞]

2 2sin cos isf t t t


Homework, Page 381Find the value of the unique real number θ between 0 and 2π that satisfies the two given conditions.

69.

If tan and sin are negative, cos must be positive. The angle must be in the fourth quadrant and the reference angle is π/4, so θ = 2π – π/4 = 7π/4

tan 1 and sin 0

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

4.4

Graphs of Sine and Cosine: Sinusoids


What you’ll learn about

The Basic Waves Sinusoids and Transformations Modeling Periodic Behavior with Sinusoids

… and why

Sine and cosine gain added significance when

used to model waves and periodic behavior.


Leading Questions

A function is a sinusoid if it can be written in the form y = a sin (bx + c) +d, where a and b ≠ 0.The function y = a cos (bx + c) +d is not a sinusoid.The amplitude of a sinusoid is |a|.The period of a sinusoid is |b|/2π.The frequency of a sinusoid is |b|/2π.Sinusoids are often used to model the behavior of periodic occurrences.

Slide 4- 21


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 3

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 FrequencyFind 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 3 3sin3

xy


Example Horizontal Stretch or Shrink and Period

Find the period of sin and use the language of 2transformations 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 compared 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 PM 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?


Following Questions

The period of the tangent function is 2π. Tangent is an odd function. Cotangent is an even function. The graph of cosecant has relative minimum

values, but no absolute minimum value. Some trig equations may be solved

algebraically. Most trig equations may be solved graphically.

Slide 4- 35


Homework

Homework Assignment #29 Read Section 4.5 Page 392, Exercises: 1 – 89 (EOO)


4.5

Graphs of Tangent, Cotangent, Secant, and Cosecant


Quick Review

2

State the period of the function.

1. cos 4

12. sin

4Find the zeros and the vertical asymptotes of the function.

13.

11

4. 2 3

5. Tell whether 4 is odd, even, or neither.

y x

y x

xy

xx

yx x

y x


Quick Review Solutions

2

State the period of the function.

1. cos 4

12. sin

4Find the zeros and the vertical asymptotes of the function.

13.

/2

8

1; 1 1

14.

2 3

5. Tell whether 4 i

1; 3, 2

y x

y x

xy

xx

yx x

y x

x

x x

s odd, even, or neither. even


What you’ll learn about

The Tangent Function The Cotangent Function The Secant Function The Cosecant Function

… and whyThis will give us functions for the remaining trigonometric ratios.


Asymptotes of the Tangent Function


Zeros of the Tangent Function


Asymptotes of the Cotangent Function


Zeros of the Cotangent Function


The Secant Function


The Cosecant Function


Basic Trigonometry Functions


Example Analyzing Trigonometric Functions

Analyze the function for domain, range, continuity, increasing or decreasing, symmetry, boundedness, extrema, asymptotes, and end behavior

secf x x


Example Transformations of Trigonometric Functions

Describe the transformations required to obtain the graph of the given function from a basic trigonometric function.

12sec

2f x x


Example Solving Trigonometric Equations

Solve the equation for x in the given interval.3sec 2, 2x x


Example Solving Trigonometric Equations With a Calculator

Solve the equation for x in the given interval.3csc 1.5, 2x x


Example Solving Trigonometric Word Problems

A hot air balloon is being blow due east from point P and traveling at a constant height of 800 ft. The angle y is formed by the ground and the line of vision from point P to the balloon. The angle changes as the balloon travels.

a. Express the horizontal distance x as a function of the angle y.

b. When the angle is , what is the horizontal distance from P?

c. An angle of is equivalent to how many degrees?

20

20

Homework, Page 381

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