DRAWING ANGLES Draw an angle with the given measure in standard position.

6. 110° 7. -10° 8. 450° 9. -900°

10. 6ir © 5TT 18 12. - 4 ? 13. 26rr

9

14. * MULTIPLE CHOICE Which angle measure is shown in the diagram?

( S ) -150° C D 210°

( g ) 570° <g) 930°

FINDING COTERMINAL ANGLES Find one positive angle and one negative angle that are coterminal with the given angle.

15. 70°

19. f

16. 255°

20. - I f

17.

21.

-125°

28ir 9

18. 820°

9 ~ 20-7T Z<J. 3

CONVERTING MEASURES Convert the degree measure to radians or the radian measure to degrees.

23540°

27. f

24. 315°

28. - ^

25. -260°

29. 5TT

26. 500°

30. 14TT 15

31. • MULTIPLE CHOICE Which angle measure is equivalent to ̂ - radians? o

( g ) 30° ( g ) 390° © 750° <g) 1110°

FINDING ARC LENGTH AND AREA Find the arc length and area of a sector with the given radius r and central angle 0.

32. r = 4 in. , 0 = - f 6 33. r = 3 m, 0 - 5TT 12

35. r = 12 ft, 0 = 150° 36. r = 18 m, 0 = 25°

38. ERROR ANALYSIS Describe and correct

34. r = 15 cm, 0 = 45°

37. r = 25 in., 0 = 270°

1, the error in finding the area of a sector A = —(6)2(40) = 720 cm2 \ / l

with a radius of 6 centimeters and a / \ central angle of 40°.

EVALUATING FUNCTIONS Evaluate the trigonometric function using a calculator if necessary. If possible, give an exact answer.

39. cos

43. cot-£-

40. sin

44. cos

41. tan

45. s i n ^

42. sec

46. esc

9

4-7T 15

I 47. CHALLENGE A rotating object that passes through an angle 0 during time t

has an angular velocity v given by the formula v = —. Find the angular

velocity of the hour hand, the minute hand, and the second hand on a 12 hour clock. Give all answers in degrees per hour.

1 3 3 Evaluate Trigonometric 1 *)•»? Functions of Any Angle .•\,

••;;?

You evaluated trigonometric functions of an acute angle. You will evaluate trigonometric functions of any angle. So you can calculate distances involving rotating objects, as in Ex. 37. __J

Key Vocabulary • unit circle • quadrantal angle • reference angle

You can generalize the right-triangle definitions of trigonometric functions from Lesson 13.1 so that they apply to any angle in standard position.

KEY CONCEPT

jndards)-

Mathematical Analysis: 9.0 Students compute, by hand, the values of the trigono-metric functions and the inverse trigonomet-ric functions at various standard points.

Mathematical Analysis: 2.0 Students know the definition of sine and cosine as y- and x-coordinates of points on the unit circle and are familiar with the graphs of the sine and cosine functions.

General Definitions of Trigonometric Functions

; ̂ Let 0 be an angle in standard position, and let (x, y) ', z be the point where the terminal side of 0 intersects '• Z the circle x2 + y2 = r2. The six trigonometric

functions of 0 are defined as follows:

r roar fifotehoQfc:

sin 0 = esc 0 -,y*o y

cos 0 = — r

tan 0 = ^ , x * 0

sec 0 = —, x # 0 x

cot 0 = - , y # 0 y

; - These functions are sometimes called circular functions.

( E X A M P L E 1 Evaluate trigonometric functions given a point

Let (—4,3) be a point on the terminal side of an angle 0 in standard position. Evaluate the six trigonometric functions of 9.

Solution

Use the Pythagorean theorem to find the value of r.

r = V*2 + / = V(-4)2 + 32 = V25 = 5 Using x = - 4 , y = 3, and r - 5, you can write the following:

sin e = - =

esc 0 = — •• y

cos 0 = — = —i r 5

sec 0 = — = -—

tan 0 = i- = -±

cot 0 = i = -±

13.2 Define General Angles and Use Radian Measure 8 6 3 8 6 6 Chapter 13 Trigonometric Ratios and Functions

KEY CONCEPT

- The Unit Circle

The circle x2 + y2 = 1, which has center (0, 0) and radius 1, is called the unit circle. The values of sin 8 and cos 8 are simply the y-coordinate and x-coordinate, respectively, of the point where the terminal side of 0 intersects the unit circle.

for Your Notebook

sin 8 y _

READING The symbol 6' is read as "theta prime.'

cos 8 = — = 4- = x r 1

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

E X A M P L E 2 Use the unit circle

:le i also the mctions

e rcle at e:

ions can nilarly.

4

Use the unit circle to evaluate the six trigonometric functions of 0 =

Solution Draw the unit circle, then draw the angle 8 = 270° in s tandard position. The terminal side of 8 intersects the unit circle at (0, -1) , so use x = 0 and y = - 1 to evaluate the trigonometric functions.

270°

s in 8 y - - 1 _ _ = - 1 esc 8 = - 1 = - 1

t a n 0 = undefined

y - i

sec 8 = — = ¥ undefined

0

(0,-1)

cot 8 = -y

o - l

aUSAlgebraj at classzone.com

GUIDED PRACTICE ] for Examples 1 and 2

Evaluate the six t r igonometric functions of 0.

-5,-12) f "

4. Use the unit circle to evaluate the six trigonometric functions of 8 = 180°

KEY CONCEPT

Reference Angle Relationships

Let 8 be an angle in standard position. The reference angle for 8 is the acute angle 6' formed by the terminal side of 6 and the x-axis. The relationship between 8 and 8' is shown below for nonquadrantal angles 8 such that 90° < 8 < 360° I 8 < 2ir).

;£> Degrees: ff = 180° - 0 Radians: ff = n - 0

Degrees: ff = 9 - 1 8 0 ° Radians: 0' = o - -n

Degrees: ff = 360° - e Radians: ff = 2n- e

E X A M P L E 3 Find reference angles

Find the reference angle 9' for (a) 0 = ~ and (b) 8 = -130°.

Solution

a. The terminal side of 8 lies in Quadrant IV. So, 0' = 2TT - — = -£ 3 3 '

b. Note that 8 is coterminal wi th 230°, whose terminal side lies in Quadrant III. So, 8' = 230° - 180° = 50°.

EVALUATING TRIGONOMETRIC FUNCTIONS Reference angles allow you to evaluate a trigonometric function for any angle ft The sign of the trigonometric function value depends on the quadrant in which 8 lies.

KEY CONCEPT

Evaluating Trigonometric Functions

Use these steps to evaluate a trigonometric function for any angle 8:

< ^ STEP 1 Find the reference angle 8'.

STEP 2 Evaluate the trigonometric function for 8'.

STEP3 Determine the sign of the trigonometric function value from the quadrant in which 01ies.

for Your Notebook

Signs of Function Values Quadrant II \y Quadrant 1

sine, esc 8: + cos 6, sec 0: — tan8,cote: -

Quadrant III sin 0, esc 0: — cos 6, sec 8: -

sin ft esc 0: cos ft sec 0: tan ft cot 0:

Quadrant IV sin ft esc 8: cos ft sec 8:

tan 8, cote : + ir tan 0, cot 6:

+ + +

X

-+ -

; ^ ,

13.3 Evaluate Trigonometric Functions of Any Angle 8 6 7 8 6 8 Chapter 13 Trigonometric Ratios and Functions

E X A M P L E 4 Use reference angles to evaluate functions

Evaluate (a) tan (-240°) and (b) c s c ^ | ^ . b

Solution

a. The angle -240° is coterminal with 120°. The reference angle is 0' = 180° - 120° = 60°. The tangent function is negative in Quadrant II, so you can write:

tan (-240°) = - t a n 60° = - V 5

b. The angle ^S is coterminal with ^ . The ° 6 o reference angle is 0' = it 5ir _ w = ^ . T h e

6 6 cosecant function is positive in Quadrant II, so you can write:

c s c ^ = cscf = 2 6 6

GUIDED PRACTICE j for Examples 3 and 4

Sketch the angle. Then find its reference angle

5. 210° 6. -260° 7. -IS. 15ir

E X A M P L E 6 Model with a trigonometric function

ROCK CLIMBING A rock climber is using a rock climbing treadmill that is 10.5 feet long. The climber begins by lying horizontally on the treadmill, which is then rotated about its midpoint by 110° so that the rock climber is climbing towards the top. If the midpoint of the treadmill is 6 feet above the ground, how high above the ground is the top of the treadmill?

Solution

sin 0 =

sin 110° =

Use definition of sine.

Substitute 110° for 0and

* ?

10.5 _ = 5.25 for r. 5.25

4.9 = y Solve for y.

• The top of the treadmill is about 6 + 4.9 = 10.9 feet above the ground.

GUIDED PRACTICE j for Examples 5 and 6

10. TRACK AND FIELD Estimate the horizontal distance traveled by a track and field long jumper who jumps at an angle of 20° and with an initial speed of 27 feet per second.

11. WHAT IF? In Example 6, how high is the top of the rock climbing treadmill if it is rotated 100° about its midpoint?

^ 0 N .

9. Evaluate cos (-210°) without using a calculator.

EXAMPLE 5 Calculate horizontal distance traveled

IDELS :ts air ,umes s >g me.

ROBOTICS The "frogbot" is a robot designed for exploring rough terrain on other planets . It can jump at a 45° angle and with an initial speed of 16 feet per second. On Earth, the horizontal distance d (in feet) traveled by a projectile launched at an angle 0 and with an initial speed v (in feet per second) is given by:

Frogbot

d = 1^ s in 20

How far can the frogbot jump on Earth?

Solution

d = | | s i n 2 #

= ^ sin (2-45°)

Write model for horizontal distance.

Substitute 16 for v and 45° for 0.

= 8 Simplify.

• The frogbot can jump a horizontal distance of 8 feet on Earth.

133 EXERCISES HOMEWORK KEY

jMMassgg^ssisBKggiftGja

O = WORKED-OUT SOLUTIONS on p. WS22 for Exs. 5,17, and 37

• = STANDARDIZED TEST PRACTICE Exs.2,11 i 37 and 39

EXAMPLE1 on p. 866 for Exs. 3-11

1. VOCABULARY Copy and complete: A(n) ? is an angle in standard position whose^erminal side lies on an axis.

2. * WRITING Given an angle 0 in Quadrant III, explain how you can use a reference angle to find cos 6.

USING A POINT Use the given point on the terminal side of an angle 0 in standard position to evaluate the six trigonometric functions of 0.

3. (8, 15) 4. (-9, 12) @ ( - 7 , -24) 6. (5, -12)

7. (2, -2) 8. ( -6 , 9) 9. ( -3 , -5 ) 10. (5, -vTT)

11. * MULTIPLE CHOICE Let (-7, -4 ) be a point on the terminal side of an angle 0 in standard position. What is the value of tan 0?

®-5 CD © f CD) 1

13.3 Evaluate Trigonometric Functions of Any Angle 8 6 9 8 7 0 Chapter 13 Trigonometric Ratios and Functions

QUADRANTAL ANGLES Evaluate the six trigonometric functions of 0.

12. 0 = 0° 13. 0 = - | 14. 0 = 540° is. e = lf

FINDING REFERENCE ANGLES Sketch the angle. Then find its reference angle.

16. -100° ( l7^150o 18. 320° 19. -370°

20. — ^ 21. i £ 3 22. 15TT 23. 13TT 6

EVALUATING FUNCTIONS Evaluate the function without using a calculator.

24. sec 135° 25. tan 240° 26. sin (-150°) 27. esc (-420°)

l l i r 28. c o s ^ 4 29. cot (--¥) 30. tan 31. sec

32. ERROR ANALYSIS Let (4,3) be a point on the terminal side of an angle 0 in standard Describe and correct the error in finding

(-¥) . position. t a n e = 2i = l \ V f igtanfl. y 3 / \ i

33. -Ar SHORT RESPONSE Write tan 0 as the ratio of two other trigonometric functions. Use this ratio to explain why tan 90° is undefined but cot 90° = 0.

34. CHALLENGE Five of the most famous numbers in mathematics — 0,1, n, e, and i — are related by the simple equation em + 1 = 0. Derive this equation using Euler's formula: ea + bl = e^cos b + i sin b).

In Exercises 35 and 36, use the formula in Example 5 on page 869.

35. FOOTBALL You and a friend each kick a football with an initial speed of 49 feet per second. Your kick is projected at an angle of 45° and your friend's kick is projected at an angle of 60°. About how much farther will your football travel than your friend's football? @HomeTutor .5 for problem solving help at classzone.com

36. IN-LINE SKATING At what speed must the in-line skater launch himself off the ramp in order to land on the other side of the ramp?

@HomeTutor ) for problem solving help at classzone.com

* SHORT RESPONSE A Ferris wheel has a radius of 75 feet. You board a car at the bottom of the Ferris wheel, which is 10 feet above the ground, and rotate 255° counterclockwise before the ride temporarily stops. How high above the ground are you when the ride stops? If the radius of the Ferris wheel is doubled, is your height above the ground doubled? Explain.

m** A Evaluate Inverse Trigonometric Functions You found values of trigonometric functions given angles. You will find angles given values of trigonometric functions. So you can find launch angles, as in Example ,4.

|P | | |abu la ry jAprsftSine

leWe cosine Vi|grse tangent

Sards

pfiUieipatical Analysis: 8.0 Students jnibw the definitions of Iffjverse trigonomet-|||§'ctlons and can

llffFthe functions.

So far in this chapter, you have learned to evaluate trigonometric functions of a given angle. In this lesson, you will study the reverse problem—finding an angle that corresponds to a given value of a trigonometric function.

Suppose you were asked to find an angle 0 whose sine is 0.5. After considering the problem, you would realize many such angles exist. For instance, the angles

ir 5v 137T 177T „ n j _7n 6 ' ~ f r ' " 6 ~ ' ~ 6 ~ , a n < 1 ~6~

all have a sine value of 0.5. To obtain a unique angle 0 such that sin 0 = 0.5, you must restrict the domain of the sine function. Domain restrictions allow the inverse sine, inverse cosine, and inverse tangent functions to be defined.

KEY CONCEPT

Inverse Trigonometric Functions

^ If - 1 < a <, 1, then the inverse sine of a is an ; ̂ angle 0, written 0 = s in - 1 a, where:

(1) sin 0 = a

Z ( 2 ) - - < 0 < - | ( o r - 9 O ° < 0 < 9 O ° )

For Voar NoUh&Qk

If - 1 < a < 1, then the inverse cosine of a is an angle 0, written 0 = cos - 1 a, where:

(1) cos 0 = a

(2) 0 < 0 < TT (or 0° < 0 < 180°)

If a is any real number, then the inverse tangent of a is an angle 0, written 0 = tan - 1 a, where:

(1) tan 0 = a

(2) — | < 0 < - | (or -90° < 0 < 90°)

13.3 Evaluate Trigonometric Functions of Any Angle 8 7 1 13.4 Evaluate Inverse Trigonometric Functions 875