Quick Review Solutions. Why 360 º ? Navigation In navigation, the course or bearing of an object is sometimes given as the angle of the line of travel.

Quick Review Solutions

1. Find the circumference of the circle with a radius of 4.5 in.

2. Find the radius of the circle with a circumference of 14 cm.

3. Given . Find if 2.2 cm and

9 in

7 / cm

4 rad 8.8ians. s r s r

cm

95.3 feet per second4. Convert 65 miles per hour into feet per second.

5. Convert 9.8 feet per second to miles per 6.681 miles hour. per hour

Why 360º?

Navigation

In navigation, the course or bearing of an object is sometimes given as the angle of the line of travel measured clockwise from due north.

Radian

A central angle of a circle has measure 1 radian if it intercepts an arc with the same length as the radius.

Example Working with Radian Measure

How many radians are in 60 degrees?

Example Working with Radian Measure

How many radians are in 60 degrees?

Since radians and 180 both measure a straight angle, use the conversion

factor radians / 180 1 to convert radians to degrees.

radians 6060 radians radians

180 180 3

Degree-Radian Conversion

180To convert radians to degrees, multiply by .

radians radians

To convert degrees to radians, multiply by .180

Arc Length Formula (Radian Measure)

If is a central angle in a circle of radius , and if is measured in

radians, then the length of the intercepted arc is given by

.

r

s

s r

Arc Length Formula (Degree Measure)

If is a central angle in a circle of radius , and if is measured in

degrees, then the length of the intercepted arc is given by

.180

r

s

rs

Example Perimeter of a Pizza Slice

Find the perimeter of a 30 slice of a large 8 in. radius pizza.

Example Perimeter of a Pizza Slice

Find the perimeter of a 30 slice of a large 8 in. radius pizza.

Let equal the arc length of the pizza's curved edge.

8 30 2404.2 in.

180 1808 in. 8 in. in.

20.2 in.

s

s

P s

P

Angular and Linear Motion

Angular speed is measured in units like revolutions per minute.

Linear speed is measured in units like miles per hour.


1. Solve for x.

x

3

2

2. Solve for x.

6

3

x

13x

3 3x

Standard Position

An acute angle θ in standard position, with one ray along the positive x-axis and the other extending into the first quadrant.

Trigonometric Functions

Let be an acute angle in the right ABC. Then

sine sin cosecant csc

cosine cos secant sec

tangent tan cotangent cot

opp hyp

hyp opp

adj hyp

hyp adj

opp adj

adj opp

Example Evaluating Trigonometric Functions of 45º

Find the values of all six trigonometric functions for an angle of 45º.



1 2 2sin 45 0.707 csc 45 1.414

2 12

1 2 2cos 45 0.707 sec 45 1.414

2 12

1 1tan 45 1 cot 45 1

1 1

opp hyp

hyp opp

adj hyp

hyp adj

opp adj

adj opp



Common Calculator Errors When Evaluating Trig Functions

Using the calculator in the wrong angle mode (degree/radians)

Using the inverse trig keys to evaluate cot, sec, and csc

Using function shorthand that the calculator does not recognize

Not closing parentheses

Example Solving a Right Triangle

A right triangle with a hypotenuse of 5 inches includes a 43 angle.

Find the measures of the other two angles and the lengths of the other

two sides.

Since it is a right triangle, one of the other angles is 90 .

That leaves 180 43 90 47 for the third angle.

Let equal the side length across from the 43 angle.

sin 47 so 3.45

Let equal

a

aa

b

the side length across from the 47 angle.

sin 47 so 3.75

bb


3 2 2 3sin 60 0.866 csc60 1.155

2 33

1 2cos60 sec60 2

2 1

3 1 3tan 60 1.732 cot 60 0.577

1 33

opp hyp

hyp opp

adj hyp

hyp adj

opp adj

adj opp


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 .

Add 360 : 45 360 405

Subtract 360 : 45 360 315

Example Finding Coterminal Angles

Find a positive angle and a negative angle that are coterminal

with .6

13Add 2 : 2

6 611

Subtract 2 : 26 6

Example Evaluating Trig Functions Determined by a Point in QI

Let be the acute angle in standard position whose terminal

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

of .

The distance from (3,5) to the origin is 34.

5 34sin 0.857 csc 1.166

534

3 34cos 0.514 sec 1.944

3345 3

tan cot3 5

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

si

P x y

r

P x y r x y

n 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.

Example Evaluating More Trig Functions

Find sin 210 without a calculator.

Example Evaluating More Trig Functions

Find sin 210 without a calculator.

An angle of 210 in standard position determines a 30 60 90 reference

triangle in the third quadrant. The lengths of the sides determines the

point ( 3, 1). The hypotenuese is 2.

sin 210 / 1/ 2

P r

y r

.

Example Using one Trig Ration to Find the Others

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

Since tan is positive the terminal side is either in QI or QIII.

The added fact that cos is negative means that the terminal

side is in QIII. Draw a reference triangle with 5, -3,

and -4.

sin -

r x

y

4 / 5 and cos -3 / 5

Unit Circle

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

Trigonometric Functions of Real Numbers

Let 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


+,+, ,

+, ,+

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 height of the wave.

f x a bx c d a

f x a bx c d a

amplitude

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.

f x a bx c d b

f x a bx c d b

period

Example Horizontal Stretch or Shrink and Period

Find the period of sin and use the language of transformations2

to describe how the graph relates to sin .

xy

y x

Example Horizontal Stretch or Shrink and Period

Find the period of sin and use the language of transformations2

to describe how the graph relates to sin .

xy

y x

2The period is 4 . The graph of sin is a horizontal

1 22

stretch of sin by a factor of 2.

xy

y x

Frequency of a Sinusoid

The of the sinusoid ( ) sin( ) is | | / 2 .

Similarly, the frequency of ( ) cos( ) is | | / 2 .

Graphically, the frequency is the number of complete cycles the

wave completes in a un

f x a bx c d b

f x a bx c d b

frequency

it interval.

Example Combining a Phase Shift with a Period Change

Construct a sinusoid with period /3 and amplitude 4 that goes through (2,0).

Example Combining a Phase Shift with a Period Change

Construct a sinusoid with period /3 and amplitude 4 that goes through (2,0).

To find the coefficient of , set 2 / | | / 3 and solve for .

Find 6. Arbitrarily choose 6.

For the amplitude set | | 4. Arbitrarily choose 4.

The graph contains (2,0) so shift the function 2

x b b

b b

a a

units to the right.

4sin(6( - 2)) 4sin(6 -12).y x x

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 sunusoid will be , and the vertical shift will be .

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

M m A

M m M mA C

p

cle of the periodic

2function. The horizontal shrink (or stretch) will be .

3. Choose an appropriate sinusoid based on behavior at some given time .

For example, at time :

( ) cos( ( - )) attai

Bp

T

T

f t A B t T C

ns a maximum value;

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

( ) sin( ( - )) is halfway between a minimum and a maximum value;

( ) - sin( ( - )) is halfway between a maximum and a minimum val

f t A B t T C

f t A B t T C

f t A B t T C

ue.

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


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


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


State the domain and range of the function.

1. ( ) -3sin 2

2. ( ) | | 2

Domain: , Range: 3,3

Domain: , Rang

3. ( ) 2cos3

e: 2,

Domain:

4. Describe t

, Range:

he behavior o

2,2

f

f x x

f x x

f x x

2

3

2

3- as .

5. Find and , given ( ) 3 and (

lim 0

3; 3

)

x

x

xy e x

f g g f f x x g x x

e

f g x g f x


Combining Trigonometric and Algebraic Functions Sums and Differences of Sinusoids Damped Oscillation

… and why

Function composition extends our ability to model

periodic phenomena like heartbeats and sound waves.

Example Combining the Cosine Function with x2

2

Graph cos and state whether the function appears to be periodic.y x


2

Graph cos and state whether the function appears to be periodic.y x

The function appears to be periodic.


2Graph cos and state whether the function appears to be periodic.y x


2Graph cos and state whether the function appears to be periodic.y x

The function appears to not be periodic.

Sums That Are Sinusoids Functions

1 1 1 2 2 2

1 2 1 1 2 2

If sin( ( )) and cos( ( )), then

y sin( ( )) cos( ( )) is a

sinusoid with period 2 / | |.

y a b x h y a b x h

y a b x h a b x h

b

Example Identifying a Sinusoid

Determine whether the following function is or is not a sinusoid.

( ) 3cos 5sinf x x x



( ) 3cos 5sinf x x x

Yes, since both functions in the sum have period 2 .



( ) cos3 sin 5f x x x



( ) cos3 sin 5f x x x

No, since cos3 has period 2 / 3 and sin 5 has period 2 / 5.x x

Damped Oscillation

The graph of ( ) cos (or ( )sin ) oscillates between the

graphs of ( ) and - ( ). When this reduces the amplitude

of the wave, it is called . The factor ( ) is called

y f x bx y f x bx

y f x y f x

f x

damped oscillation

the .damping factor

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

4.7

Inverse Trigonometric Functions

Quick Review

State the sign (positive or negative) of the sine, cosine, and tangent

in quadrant

1. I

2. III

Find the exact value.

3. cos 64

4. tan 311

5. sin 6


State the sign (positive or negative) of the sine, cosine, and tangent

in quadrant

1. I

2. III

Find the exact value.

3. cos 64

4. tan 311

5. si

+,+,+

,

n

,+

3 / 2

3

1/ 2 6


Inverse Sine Function Inverse Cosine and Tangent Functions Composing Trigonometric and Inverse Trigonometric

Functions Applications of Inverse Trigonometric Functions

… and whyInverse trig functions can be used to solve trigonometric equations.

Inverse Sine Function

Inverse Sine Function (Arcsine Function)

-1

The unique angle in the interval - / 2, / 2 such that sin is the

(or ) of , denoted or .

The domain of sin is [-1,1] and the range is - / 2, / 2 .

y y x

x

y x

-1inverse sine arcsine sin arcsin x x

Example Evaluate sin-1x Without a Calculator

-1 1Find the exact value without a calculator: sin

2


-1 1Find the exact value without a calculator: sin

2

Find the point on the right half of the unit circle whose -coordinate is -1/ 2

and draw a reference triangle. Recognize this as a special ratio, and the

angle in the interval [- / 2, / 2] whose sin is -

y

1/ 2 is - / 6.


-1Find the exact value without a calculator: sin sin .10


-1Find the exact value without a calculator: sin sin .10

-1

Draw an angle /10 in standard position and mark its -coordinate on

the -axis. The angle in the interval [- /2, /2] whose sine is this

number is /10. Therefore, sin sin .10 10

y

y

Inverse Cosine (Arccosine Function)

Inverse Cosine (Arccosine Function)

-1

The unique angle in the interval 0, such that cos is the


The domain of cos is [-1,1] and the range is 0, .

y y x

x

y x

-1inverse cosine arccosine cos arccos x x

Inverse Tangent Function (Arctangent Function)

Inverse Tangent Function (Arctangent Function)

-1

The unique angle in the interval ( - / 2, / 2) such that tan is the


The domain of tan is (- , ) and the range is ( - / 2, / 2).

y y x

x

y x

-1inverse tangent arctangent tan arctan x x

End Behavior of the Tangent Function

Composing Trigonometric and Inverse Trigonometric Functions

-1 -1 -1

The following equations are always true whenever they are defined:

sin sin cos cos tan tan

The following equations are only true for values in the "restricted"

domains of si

x x x x x x

x

1 -1 -1

n, cos, and tan:

sin sin cos cos tan tanx x x x x x

Example Composing Trig Functions with Arcsine

-1Compose each of the six basic trig functions with sin and reduce the

composite function to an algebraic expression involving no trig functions.

x

Example Composing Trig Functions with Arcsine

-1Compose each of the six basic trig functions with sin and reduce the

composite function to an algebraic expression involving no trig functions.

x

-1 -1

-1 2 -1

2

2

-1 -1

2

Use the triangle to find the required ratios:

1sin(sin )) csc(sin ))

1cos(sin )) 1 sec(sin ))

1

1tan(sin )) cot(sin ))

1

x x xx

x x xx

x xx x

xx

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

4.8

Solving Problems with Trigonometry

Quick Review

1. Solve for a.

a

3

23º

Quick Review

2. Find the complement of 47 .

3. Find the supplement of 47 .

4. State the bearing that describes the direction NW (northwest).

5. State the amplitude and period of the sinusoid 3cos 2( 1).x


1. Solve for a.

a

3

23º 7.678


2. Find the complement of 47 .

3. Find the supplement of 47 .

4. State the bearing that

43

133

1describes the direction NW (northwest).

5. State the amplitude and period of the sinus

35

oid 3c

os 2( 1).

3, A p

x


More Right Triangle Problems Simple Harmonic Motion

… and why

These problems illustrate some of the better-

known applications of trigonometry.

Angle of Elevation, Angle of Depression

An angle of elevation is the angle through which the eye moves up from horizontal to look at something above. An angle of depression is the angle through which the eye moves down from horizontal to look at something below.

Example Using Angle of Elevation

The angle of elevation from the buoy to the top of the Barnegat Bay lighthouse 130 feet above the surface of the water is 5º. Find the distance x from the base of the lighthouse to the buoy.

130

x

5º

Example Using Angle of Elevation

The angle of elevation from the buoy to the top of the Barnegat Bay lighthouse 130 feet above the surface of the water is 5º. Find the distance x from the base of the lighthouse to the buoy.

130

x

5º

130tan 5

1301485.9

tan 5The buoy is about 1486 feet from the base of the lighthouse.

x

x

Simple Harmonic Motion

A point moving on a number line is in if its

directed distance from the origin is given by either

sin or cos , where and are real numbers and 0.

The motion has

d

d a t d a t a

simple harmonic motion

frequency / 2 , which is the number of

oscillations per unit of time.

Example Calculating Harmonic Motion

A mass oscillating up and down on the bottom of a spring (assuming perfect elasticity and no friction or air resistance) can be modeled as harmonic motion. If the weight is displaced a maximum of 4 cm, find the modeling equation if it takes 3 seconds to complete one cycle.

Example Calculating Harmonic Motion

A mass oscillating up and down on the bottom of a spring (assuming perfect elasticity and no friction or air resistance) can be modeled as harmonic motion. If the weight is displaced a maximum of 4 cm, find the modeling equation if it takes 3 seconds to complete one cycle.

Assume the spring is at the origin of the coordinate system when 0 and

use the equation sin .

The maximum displacement is 4 cm, so 4.

One cycle takes 3 sec, so the period is 3 and the frequency i

t

d a t

a

s 1/3.

1 2Therefore, and .

2 3 32

Put this together and sin 4sin .3

d a t d t

Chapter Test

1. The point (-1, 3) is on the terminal side of an angle in standard position.

Give the smallest positive angle measure in both degrees and radians.

2. Evaluate sec without using a calculator.3

3

. Find all six trigonometric functions of in ABC.

5

C12

αA

B

Chapter Test

4. The point (-5,-3) is on the terminal side of angle . Evaluate

the six trigonometric functions for .

5. Use transformations to describe how the graph of the function

-2 -3sin - is related to the fy x

unction sin . Graph two periods.

6. State the amplitude, period, phase shift, domain and range for

( ) 1.5sin 2 - / 4 .

y x

f x x

Chapter Test

2

-1

7. Find the exact value of without using a calculator:

tan -1, 0 .

sin8. Describe the end behavior of ( ) .

9. Find an algebraic expression equivalent to tan cos .

10. From the top of a 150-

x

x x

xf x

xx

ft building Kana observes a car

moving toward her. If the angle of depression of the car

changes from 18 to 42 during the observation, how far

does the car travel?

1. The point (-1, 3) is on the terminal side of an angle in standard position.

Give the smallest posit

120

ive angle measure in both degrees and radians.

2. Evaluate s

2 / 3 radian

ec without us3

s

2

sin

ing a calculator.

3. Find all six trigonometric functions of in ABC.

5 /13 csc =13/5

cos =12/13 sec =13/12

tan 5 /12 cot =12/5

Chapter Test Solutions

5

C 12

αA

B

4. The point (-5,-3) is on the terminal side of angle . Evaluate

the six trigonometric functions for sin 3/ 34; csc = 34/3;

cos = 5/ 34; sec = 34/5; tan 3/ 5; cot

.

5. Use transformations

=5/3

to

describe how the graph of the function

-2 -3sin - is related to the function sin . Graph two peri

translation right units, vertical stretch by a factor of 3, reflected

across the

ods.

-axis, tranx

y x y x

6. State the amplitude, period, phase shift, domain and range for

( ) 1.5sin 2 - / 4 .

slation down 2 units.

A=1.5; p= ; ps= /8; domain: - , ; range: -1.5,1.5f x x



2

-1

7. Find the exact value of without using a calculator:

tan -1, 0 .

sin8. Describe the end behavior of ( ) .

9. Find an algebraic expression

3 /4

As | | , ( ) 0

equivalent to tan cos .-

1

x

x x

xf x

x f x

x

x

x

2

10. From the top of a 150-ft building Kana observes a car

moving toward her. If the angle of depression of the car

changes from 18 to 42 during the observation, how far

does the car travel? 1 0 c 5 (

x

ot18 cot 42 ) 295 ft

Quick Review Solutions. Why 360 º ? Navigation In navigation, the course or bearing of an object is sometimes given as the angle of the line of travel.

Documents

trigonometric functions

trig functions slide

x slide

qi slide

negative angle slide

right triangle slide

pizza slice slide

degreeradian conversion