QUADRANTAL ANGLES Evaluate the six trigonometric functions of 0.

12. 0 = 0° 13. 0 = 14. 6 = 540c 15. e 7ir 2

FINDING REFERENCE ANGLES Sketch the angle. Then find its reference angle. 16. -100° (l7^ 150° 18. 320° 19. -370°

20. - 5ir 21. 8TT 22. 15TT 4 23. - 13w

EVALUATING FUNCTIONS Evaluate the function without using a calculator. 24. sec 135° 25. tan 240° 26. sin (-150°)

28. c o s ^ 4 29. cot ( • - f ) 30. tan (-¥) 27. esc (-420°)

31. s e c ^ o

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

tan 6 = £ = 4 X J 33. • 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, IT, e, and i — are related by the simple equation em + 1 = 0. Derive this equation using Euler's formula: e" + b' = e°(cos b + i sin b).

ROBLEM SOLVING

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? @HomeTtltorj 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 j for problem solving help at classzone.com

I (37J • 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.

3 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 o c , ||l|fcsine

^i^ifse cosine ftpfce tangent

Sards

MgilfemaUcal Analysis: 8.0 Students ip?ir the definitions of |llnverse trigonomet-nc'functions and can l l m the 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

^ ^ J L M . i Z E . a n d - ^ 6 6 6 6 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

-E If -1 < a < 1, then the inverse sine of a is an " ^ angle 0, written 0 = sin-1 a, where: ! •• (1) sin 6 = a

(2) - - < 6 < j (or -90° < 6 < 90°)

If -1 < a < 1, then the inverse cosine of a is an ; - angle 0, written 8 = cos-1 a, where:

'.' (1) cos 6 = a ; ^ (2) 0 < 6 < IT (or 0° < 6 < 180°)

For Your Notebook

]'. 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) ~ J < 0 < ± (or -90° < 0 < 90°)

77

K ^ « ."'- j

2 , ,

X

13.3 Evaluate Trigonometric Functions of Any Angle 871 13.4 Evaluate Inverse Trigonometric Functions 875

f E X A M P L E 1 Evaluate inverse trigonometric functions

Evaluate the expression in both radians and degrees.

.-1V3 a. cos

Solution

b. sin 12 c. t a n - 1 (-V3)

V3 a. When 0 < 0 < TT, or 0° < 0 < 180°, the angle whose cosine is -y - is:

c - l V3 _ IT -iV3 I = cos"' 1± = £ or 0 = cos"1 If = 30°

b. There is no angle whose sine is 2. So, sin ' 2 is undefined.

c. When -— < 0 < ~, or -90° < 0 < 90°, the angle whose tangent is - V3 is:

0 - tan" 1 (-V5) = - f or 0 = tan" 1 (-V5) = -60°

E X A M P L E 2 Solve a trigonometric equation

USE A CALCULATOR On most calculators, you can evaluate inverse trigonometric functions using the keys | S 3 U S for inverse sine, jj§jj §55 for inverse cosine, and jffEii B 3 f ° r inverse tangent.

Solve the equation sin 0 = — | where 180° < 0 < 270°

Solution JIBP T Use a calculator to determine that in the

interval -90° < 0 < 90°, the angle whose

sine is — | is s in - 1 (--1) - -38.7°. This

angle is in Quadrant IV, as shown.

STEP 2 Find the angle in Quadrant HI (where 180° < 0 < 270°) that has the same sine value as the angle in Step 1. The angle is:

0 = 180° + 38.7° = 218.7°

CHECK Use a calculator to check the answer.

sin 218.7° •• -0.625= —£•/

:s hoices

cthat sset , not

E X A M P L E 3 Standardized Test Practice

What is the measure of the angle 0 in the triangle shown?

(5 ) 28.6° CD 33.1° 6

" © 56.9° CD 61.4°

Solution In the right triangle, you are given the lengths of the side adjacent to 0 and the hypotenuse, so use the inverse cosine function to solve for 0.

cos 0 adj hyp

= ~ - ,'ivv£KBSSaaBMff^ 0 = COS - i J i -l l 56.9°

• The correct answer is C. <3) <S> ® <B>

EXAMPLE 4 Write and solve a trigonometric equation

MONSTER TRUCKS A monster truck drives off a ramp in order to jump onto a row of cars. The ramp has a height of 8 feet and a horizontal length of 20 feet. What is the angle 0 of the ramp?

Solution STiP 1 Draw a triangle that represents the ramp.

STEP 2 Write a trigonometric equation that involves the ratio of the ramp's height and horizontal length.

t a n 0 = ^ = A adj 20 STEP 3 Use a calculator to find the measure of 0.

= tan - l 8 _ 20

« 21.8°

• The angle of the ramp is about 22°.

20 ft

GUIDED PRACTICE 1 for Examples 1 and 2

Evaluate the expression in both radians and degrees.

1. sin" V2 2. cos 1 1

Solve the equation for 9.

5. cos 0 = 0.4; 270° < 0 < 360°

7. sin 0 = - 0 . 2 3 ; 270° < 0 < 360°

9. sin 0 = 0.62; 90° < 0 < 180°

876 Chapter 13 Trigonometric Ratios and Functions

3. t a n - 1 (-1) 4. sin 1\—x

6. t an0=2.1 ;18O°<0<27O°

8. tan 0 = 4.7; 180° < 0 < 270°

10. cos 0 = -0 .39; 180° < 0 < 270°

.# GUIDED PRACTICE j for Examples 3 and 4

Find the measure of the angle 0.

11.

14. WHAT IF? In Example 4, suppose a monster truck drives 26 feet on a ramp before jumping onto a row of cars. If the r amp is 10 feet high, what is the angle 0 of the ramp?

13.4 Evaluate Inverse Trigonometric Functions 877

EXAMPLE 1 on p. 876 for Exs. 3-11

SKILL PRACTICE

1. VOCABULARY Copy and complete: The _L_ sine of ^ is —-, or 30°.

2. * WRITING Explain why t an - 1 3 is defined, but cos - 1 3 is undefined.

EVALUATING EXPRESSIONS Evaluate the expression without using a calculator. Give your answer in both radians and degrees.

• - 1 1 4. t an - 1 (-1) 5. cos - 1 0 6. cos - 1 (-2)

EXAMPLE 2 on p. 876 for Exs. 20-26

EXAMPLE 3 on p. 877 for Exs. 27-29

3. sin

^ 7 ^ s i n _ 1 V3 8. s i n - 1 ^ 9. tan" 3 10. cos -(4 11. • MULTIPLE CHOICE What is the value of the expression cos x-y-?

(A) 0° CD 30° (g) 45° (g> 60°

USING A CALCULATOR Use a calculator to evaluate the expression in both radians and degrees.

12. s in - 1 0.18 13. t an - 1 2.6 14. cos - 1 0.36 15. cos - 1 (-0.4)

16. t an - 1 (-0.75) 17. s in - 1 (-0.2) 18. s in - 1 0 .8 19. cos-10.99

SOLVING EQUATIONS Solve the equation for 0.

20. cos 0 = -0.82; 180° < 0 < 270° 21. sin 9 = -0.45; 180° < 9 < 270°

22. sin 6 = 0.15; 90° < 9 < 180° (23^ tan 9 = 3.2; 180° < 9 < 270°

24. tan 9 = - 5 . 3 ; 90° < 9 < 180° 25. cos 9 = 0.25; 270° < 9 < 360°

26. ERROR ANALYSIS Describe and correct the error in solving the equation sin 9 = 0.7 where 90° < 9 < 180°.

The angle whose sine is 0.7 is sin- 1 0.7 «• 44.4°. so 0 « 44.4°, X

FINDING ANGLES Find the measure of the angle 0.

27. jf\ 28. 6

iT 29.

30. • OPEN-ENDED MATH Suppose cos 9 > 0 and sin 9 < 0. Give a possible value of 9 such that -360° < 9 < 0°.

31. • OPEN-ENDED MATH Suppose sin 9 < 0 and tan 9 > 0. Give a possible value of 9 such that 360° < 9 < 720°.

CHALLENGE Rewrite the expression so that it does not involve trigonometric functions or inverse trigonometric functions.

32. csc (sin-1 x) 33. cot ( tan - 1 x) 34. sec (cos- 1 x)

16-1 Angles and the Unit Circle Consider a circle, centered a t the origin, with two rays extending

from the center as shown. One ray is fixed along the positive x-axis. The other ray can rotate about the center.

These rays form an angle. The fixed ray is called the init ial side of the angle. The other is called the terminal side of the angle.

terminal side An angle with its vertex at the origin and its initial side along the positive x-axis is said to be in standard position.

initial side

Start wi th both sides along the positive x-axis. As the terminal side is rotated counterclockwise, the measure of the angle formed increases.

30 degrees 150 degrees

The rotation of the terminal side of the angle may include one or more complete revolutions about the center. The measurement of an angle representing one complete revolution of the circle is 360 de-grees, usually writ ten 360°.

one revolution 360°

two revolutions 360° x 2 or 720°

Angles t ha t differ by one or more complete rotations of the circle are called c o t e r m i n a l ang l e s . For example, 74°, 434°, and 794° are coterminal angles.

210 degrees

The most widely used unit of angle measure is the degree.

three revolutions 360° x 3 or 1,080°

8 7 8 Chapter 13 Trigonometric Ratios and Functions Chapter 16 479