8.3 Trigonometric Functions of Any Angle

Trigonometric Functions of

Any Angle

8.3

Essential Question

How can you use the unit circle to define the

trigonometric functions of any angle?

Let ¦È be an angle in standard position with (x, y) a point on the terminal side of ¦È and

¡ª

r = ¡Ì x2 + y2 ¡Ù 0. The six trigonometric functions of ¦È are defined as shown.

y

sin ¦È = ¡ª

r

r

csc ¦È = ¡ª, y ¡Ù 0

y

x

cos ¦È = ¡ª

r

r

sec ¦È = ¡ª, x ¡Ù 0

x

y

tan ¦È = ¡ª, x ¡Ù 0

x

x

cot ¦È = ¡ª, y ¡Ù 0

y

y

(x, y)

r

¦È

x

Writing Trigonometric Functions

Work with a partner. Find the sine, cosine, and tangent of the angle ¦È in standard

position whose terminal side intersects the unit circle at the point (x, y) shown.

a.

(

?1 , 3

2 2

(

y

b.

(?12 , 12 (

c.

y

x

y

x

x

(0, ?1)

d.

e.

y

f.

y

y

(?1, 0)

x

x

x

( 12 , ? 2 3 (

( 12 , ?12 (

CONSTRUCTING

VIABLE ARGUMENTS

To be proficient in

math, you need to

understand and use stated

assumptions, definitions,

and previously established

results.

Communicate Your Answer

2. How can you use the unit circle to define the trigonometric functions of any angle?

3. For which angles are each function undefined? Explain your reasoning.

a. tangent

b. cotangent

Section 8.3

Int_Math3_PE_08.03.indd 425

c. secant

d. cosecant

Trigonometric Functions of Any Angle

425

1/30/15 4:48 PM

8.3 Lesson

What You Will Learn

Evaluate trigonometric functions of any angle.

Find and use reference angles to evaluate trigonometric functions.

Core Vocabul

Vocabulary

larry

unit circle, p. 427

quadrantal angle, p. 427

reference angle, p. 428

Previous

circle

radius

Pythagorean Theorem

Trigonometric Functions of Any Angle

You can generalize the right-triangle definitions of trigonometric functions so that they

apply to any angle in standard position.

Core Concept

General Definitions of Trigonometric Functions

Let ¦È be an angle in standard position, and let (x, y)

be the point where the terminal side of ¦È intersects

the circle x2 + y2 = r2. The six trigonometric

functions of ¦È are defined as shown.

y

sin ¦È = ¡ª

r

x

cos ¦È = ¡ª

r

y

tan ¦È = ¡ª, x ¡Ù 0

x

y

¦È

(x, y)

r

r

csc ¦È = ¡ª, y ¡Ù 0

y

r

sec ¦È = ¡ª, x ¡Ù 0

x

x

cot ¦È = ¡ª, y ¡Ù 0

y

x

These functions are sometimes called circular functions.

Evaluating Trigonometric Functions Given a Point

Let (?4, 3) be a point on the terminal side of

an angle ¦È in standard position. Evaluate the

six trigonometric functions of ¦È.

y

¦È

(?4, 3)

SOLUTION

r

Use the Pythagorean Theorem to find the length of r.

x

¡ª

r = ¡Ì x2 + y2

¡ª

= ¡Ì (?4)2 + 32

¡ª

= ¡Ì 25

=5

Using x = ?4, y = 3, and r = 5, the values of the six trigonometric functions of ¦È are:

426

Chapter 8

Int_Math3_PE_08.03.indd 426

y 3

sin ¦È = ¡ª = ¡ª

r 5

r 5

csc ¦È = ¡ª = ¡ª

y 3

4

x

cos ¦È = ¡ª = ?¡ª

r

5

5

r

sec ¦È = ¡ª = ?¡ª

x

4

3

y

tan ¦È = ¡ª = ?¡ª

x

4

4

x

cot ¦È = ¡ª = ?¡ª

y

3

Trigonometric Ratios and Functions

1/30/15 4:48 PM

Core Concept

The Unit Circle

y

The circle + = 1, which has center (0, 0)

and radius 1, is called the unit circle. The values

of sin ¦È and cos ¦È are simply the y-coordinate and

x-coordinate, respectively, of the point where the

terminal side of ¦È intersects the unit circle.

x2

y ?r

sin ¦È = ¡ª = ¡ª = ?1.

r

r

The other functions can be

evaluated similarly.

¦È

x

r=1

y y

sin ¦È = ¡ª = ¡ª = y

r 1

x x

cos ¦È = ¡ª = ¡ª = x

r 1

ANOTHER WAY

The general circle

x2 + y2 = r2 can also be used

to find the six trigonometric

functions of ¦È. The terminal

side of ¦È intersects the circle

at (0, ?r). So,

y2

(x, y)

It is convenient to use the unit circle to find trigonometric functions of quadrantal

angles. A quadrantal angle is an angle in standard position whose terminal side lies on

¦Ð

an axis. The measure of a quadrantal angle is always a multiple of 90?, or ¡ª radians.

2

Using the Unit Circle

Use the unit circle to evaluate the six trigonometric functions of ¦È = 270?.

SOLUTION

y

Step 1 Draw a unit circle with the angle ¦È = 270? in

standard position.

¦È

Step 2 Identify the point where the terminal side

of ¦È intersects the unit circle. The terminal

side of ¦È intersects the unit circle at (0, ?1).

x

Step 3 Find the values of the six trigonometric

functions. Let x = 0 and y = ?1 to evaluate

the trigonometric functions.

(0, ?1)

y ?1

sin ¦È = ¡ª = ¡ª = ?1

r

1

1

r

csc ¦È = ¡ª = ¡ª = ?1

y ?1

x 0

cos ¦È = ¡ª = ¡ª = 0

r 1

r 1

sec ¦È = ¡ª = ¡ª

x 0

y ?1

tan ¦È = ¡ª = ¡ª

x

0

0

x

cot ¦È = ¡ª = ¡ª = 0

y ?1

undefined

Monitoring Progress

undefined

Help in English and Spanish at

Evaluate the six trigonometric functions of ¦È.

1.

2.

y

(?8, 15)

3.

y

¦È

¦È

x

(3, ?3)

y

¦È

x

x

(?5, ?12)

4. Use the unit circle to evaluate the six trigonometric functions of ¦È = 180?.

Section 8.3

Int_Math3_PE_08.03.indd 427

Trigonometric Functions of Any Angle

427

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Reference Angles

Core Concept

READING

Reference Angle Relationships

The symbol ¦È¡ä is read as

¡°theta prime.¡±

Let ¦È be an angle in standard position. The reference angle for ¦È is the acute

angle ¦È¡ä formed by the terminal side of ¦È and the x-axis. The relationship between

¦È and ¦È¡ä is shown below for nonquadrantal angles ¦È such that 90¡ã < ¦È < 360¡ã or,

¦Ð

in radians, ¡ª < ¦È < 2¦Ð.

2

y

¦È¡ä

Quadrant IV

Quadrant III

Quadrant II

y

y

¦È

¦È

¦È

x

Degrees: ¦È ¡ä = 180¡ã ? ¦È

Radians: ¦È ¡ä = ¦Ð ? ¦È

¦È¡ä

x

Degrees: ¦È ¡ä = ¦È ? 180¡ã

Radians: ¦È ¡ä = ¦È ? ¦Ð

¦È¡ä

x

Degrees: ¦È ¡ä = 360¡ã ? ¦È

Radians: ¦È ¡ä = 2¦Ð ? ¦È

Finding Reference Angles

5¦Ð

Find the reference angle ¦È ¡ä for (a) ¦È = ¡ª and (b) ¦È = ?130?.

3

SOLUTION

a. The terminal side of ¦È lies in Quadrant IV. So,

y

x

¦È¡ä

¦È

y

5¦Ð ¦Ð

¦È¡ä = 2¦Ð ? ¡ª = ¡ª. The figure at the right shows

3

3

¦Ð

5¦Ð

¦È = ¡ª and ¦È¡ä = ¡ª.

3

3

b. Note that ¦È is coterminal with 230?, whose terminal side

lies in Quadrant III. So, ¦È¡ä = 230? ? 180? = 50?. The

figure at the left shows ¦È = ?130? and ¦È¡ä = 50?.

¦È

x

¦È¡ä

Reference angles allow you to evaluate a trigonometric function for any angle ¦È. The

sign of the trigonometric function value depends on the quadrant in which ¦È lies.

Core Concept

Evaluating Trigonometric Functions

Use these steps to evaluate a

trigonometric function for any angle ¦È:

Step 1 Find the reference angle ¦È¡ä.

Step 2 Evaluate the trigonometric

function for ¦È¡ä.

Step 3 Determine the sign of the

trigonometric function value

from the quadrant in which

¦È lies.

428

Chapter 8

Int_Math3_PE_08.03.indd 428

Signs of Function Values

Quadrant II

sin ¦È, csc ¦È : +

cos ¦È , sec ¦È : ?

tan ¦È , cot ¦È : ?

Quadrant III

sin ¦È, csc ¦È : ?

cos ¦È , sec ¦È : ?

tan ¦È , cot ¦È : +

Quadrant I

sin ¦È, csc ¦È : +

cos ¦È , sec ¦È : +

tan ¦È , cot ¦È : +

y

Quadrant IV x

sin ¦È, csc ¦È : ?

cos ¦È , sec ¦È : +

tan ¦È , cot ¦È : ?

Trigonometric Ratios and Functions

1/30/15 4:49 PM

Using Reference Angles to Evaluate Functions

17¦Ð

Evaluate (a) tan(?240?) and (b) csc ¡ª.

6

SOLUTION

y

a. The angle ?240? is coterminal with 120?. The reference

angle is ¦È¡ä = 180? ? 120? = 60?. The tangent function ¦È¡ä = 60¡ã

is negative in Quadrant II, so

¡ª

x

tan(?240?) = ?tan 60? = ?¡Ì 3 .

¦È = ?240¡ã

5¦Ð

17¦Ð

b. The angle ¡ª is coterminal with ¡ª. The

6

6

reference angle is

y

5¦Ð ¦Ð

¦È¡ä = ¦Ð ? ¡ª = ¡ª.

6

6

The cosecant function is positive in Quadrant II, so

INTERPRETING

MODELS

This model neglects air

resistance and assumes

that the projectile¡¯s

starting and ending

heights are the same.

¦È¡ä= ¦Ð6

17¦Ð

¦È=

6

17¦Ð

¦Ð

csc ¡ª = csc ¡ª = 2.

6

6

x

Solving a Real-Life Problem

The horizontal distance d (in feet) traveled by a projectile launched at

an angle ¦È and with an initial speed v (in feet per second) is given by

v2

d = ¡ª sin 2¦È.

Model for horizontal distance

32

Estimate the horizontal distance traveled by a golf ball

that is hit at an angle of 50¡ã with an initial speed of

105 feet per second.

50¡ã

SOLUTION

Note that the golf ball is launched at an angle of ¦È = 50? with initial speed

of v = 105 feet per second.

v2

d = ¡ª sin 2¦È

32

1052

= ¡ª sin(2 50¡ã)

32

Write model for horizontal distance.

?

Substitute 105 for v and 50? for ¦È.

¡Ö 339

Use a calculator.

The golf ball travels a horizontal distance of about 339 feet.

Monitoring Progress

Help in English and Spanish at

Sketch the angle. Then find its reference angle.

?7¦Ð

9

Evaluate the function without using a calculator.

6. ?260¡ã

5. 210¡ã

7. ¡ª

15¦Ð

4

8. ¡ª

11¦Ð

4

11. Use the model given in Example 5 to estimate the horizontal distance traveled

by a track and field long jumper who jumps at an angle of 20¡ã and with an initial

speed of 27 feet per second.

9. cos(?210?)

Section 8.3

Int_Math3_PE_08.03.indd 429

10. sec ¡ª

Trigonometric Functions of Any Angle

429

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