3 ^ Define General Angles ^ • ^ and Use Radian Measure

You used acute angles measured in degrees. You will use general angles that may be measured in radians. So you can find the area of a curved playing field, as in Example 4.

/ocabulary filial side

linal side ftndard position iferminal

ibthematkal ifalysis: 1.0 Students nderstand the notion f angle and how to leasure I t In both de-rees and radians. They an convert between egrees and radians.

In Lesson 13.1, you worked only with acute angles. In this lesson, you will study angles with measures that can be any real numbers.

KEY CONCEPT

Angles in Standard Position In a coordinate plane, an angle can be formed by fixing one ray, called the initial side, and rotating the other ray, called the terminal side, about the vertex.

An angle is in standard position if its vertex is at the origin and its initial side lies on the positive x-axis.

For foar Notebook

180°

270°

The measure of an angle is positive if the rotation of its terminal side is counterclockwise, and negative if the rotation is clockwise. The terminal side of an angle can make more than one complete rotation.

COTERMINAL ANCLES In Example 1, the angles 500° and 140° are coterminal because their terminal sides coincide. An angle coterminal with a given angle can be found by adding or subtracting multiples of 360°.

(jEXAMPLE 2 Find coterminal angles

s

Find one positive angle and one negative angle that are coterminal with (a) -45° and (b) 395°.

Solution

There are many such angles, depending on what multiple of 360° is added or subtracted.

a. -45° + 360° = 315° -45° - 360° = -405°

b. 395° - 360° = 35° 395° - 2(360°) = -325°

GUIDED PRACTICE j for Examples 1 and 2

Draw an angle with the given measure in standard position. Then find one positive coterminal angle and one negative coterminal angle.

1. 65° 2. 230° 3. 300° 4. 740°

EXAMPLE 1 Draw angles in standard position

Draw an angle with the given measure in standard position,

a. 240° b. 500° c. -50°

Solution

Because 240° is 60° more than 180°, the terminal side is 60° counterclockwise past the negative x-axis.

Because 500° is 140° more than 360°, the terminal side makes one whole revolution counterclockwise plus 140° more.

V 1

• " f I V •

•y 1—-^140°

J ' ^ 5 0 0 °

1

Because —50° is negative, the terminal side is 50° clockwise from the positive x-axis.

RADIAN MEASURE Angles can also be measured in radians. To define a radian, consider a circle with radius r centered at the origin as shown. One radian is the measure of an angle in standard position whose terminal side intercepts an arc of length r.

Because the circumference of a circle is 27rr, there are 27r radians in a full circle. Degree measure and radian measure are therefore related by the equation 360° = 2TT radians, or 180° = IT radiant.

KEY CONCEPT For Your Notebook Converting Between Degrees and Radians Degrees to radians Radians to degrees

: b y

Multiply degree measure IT radians

Multiply radian measure 180°

180° by 7r radians'

13.2 Define General Angles and Use Radian Measure 8 5 9 360 Chapter 13 Trigonometric Ratios and Functions

E X A M P L E 3 Convert between degrees and radians

HNC nit "radians" is omitted. For ice, the measure radians may be

in simply as —^-.

Convert (a) 125° to radians and (b) — ~ radians to degrees.

a. 125° , „co/ IT radians \ 1 2 5 I 180° )

= ^ radians 36

b. - £ = (-JLradiansV 18°° ) 12 \ 12 A 7T radians/

-15°

The diagram shovys equivalent degree and radian measures for special angles from 0° to 360° (0 radians to 1-n radians). You may find it helpful to memorize the equivalent degree and radian measures of special angles in the first quadrant and for 90° = ~ radians. All other special angles

• ^ are just multiples of these angles.

f E X A M P L E 4 Solve a multi-step problem

AVOID ERRORS You must write the measure of an angle in radians when using the formulas for the arc length and area of a sector.

SOFTBALL A softball field forms a sector with the dimensions shown. Find the length of the outfield fence and the area of the field.

Solution

STEP 1 Convert the measure of the central angle to radians.

90° = 9 o° (^ radians \ = ^ d i I 180° I 2

STEP 2 Find the arc length and the area of the sector.

Arc length: s = rd= 180(-|) = 90TT «- 283 feet

Area: ,4 = ^ 0 = | (180)2/- | ) = 8100-n- - 25,400 ft2

• The length of the outfield fence is about 283 feet. The area of the field is about 25,400 square feet.

I GUIDED PRACTICE for Example 4

9. WHAT IF? In Example 4, estimate the length of the outfield fence and the area of the field if the outfield fence is 220 feet from home plate.

GUIDED PRACTICE ] for Example 3

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

5. 135° 6. -50° 577 8. 10

SECTORS OF CIRCLES A sector is a region of a circle that is bounded by two radii and an arc of the circle. The central angle 6 of a sector is the angle formed by the two radii. There are simple formulas for the arc length and area of a sector when the central angle is measured in radians.

KEY CONCEPT

Arc Length and Area of a Sector

The arc length s and area A of a sector with radius r and central angle 6 (measured in radians) are as follows.

Arc length: s = rd

Area: A = -1*6

For Your Notebook

ength

13.2 EXERCISES HOMEWORK j Q _ WORKED-OUT SOLUTIONS KEY; on p. WS22 for Exs. 11,23, and 51

! • = STANDARDIZED TEST PRACTICE EXS. 2,14,31,50, and 53

SKILL PRACTICE

1. VOCABULARY Copy and complete: An angle is in standard position if its vertex is at the ? and its ? lies on the positive jc-axis.

2. * WRITING How does the sign of an angle's measure determine its direction of rotation?

EXAMPLES 1 and 3 on pp. 859-861 for Exs. 3-14

VISUAL THINKING Match the angle measure wijh the angle.

3. -240° 4. 600° 5.

A. | y B. V *y C.

13.2 Define General Angles and Use Radian Measure 8 6 1 8 6 2 Chapter 13 Trigonometric Ratios and Functions

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

6. 110° 7. -10° 8. 450° 9. -900°

10. 6TT ©S 12. - 5TT 13. 26TT

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

(A) -150° <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 7TT 6

17. -125°

21. 28^

18. 820°

22. ^

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

23^40°

27. f

24. 315°

28. - • ?

25. -260°

29 . Sir

26. 500°

30. 14TT 15

13ir, 31. * MULTIPLE CHOICE Which angle measure is equivalent to -=^=. radians?

(£> 30° (g ) 390° (g) 750° Cg) 1110°

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

32. r = 4in., 0 = -£ 33. r = 3 m , 0 = 4 ?

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°

the error in finding the area of a sector * — ' "=->2<,x'v> — nnr, —2 with a radius of 6 centimeters and a central angle of 40°.

A = i ( 6 ) " ( 4 0 ) = 720 c m ' x: EVALUATING FUNCTIONS Evaluate the trigonometric function using a calculator if necessary. If possible, give an exact answer.

39. cos-^-

43. cot 4

40. sin

44. cos

41. tan-£-6 42. sec —

45. sin 3TT 46. esc 4ir 15

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

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.

13 3 E v a , u a t e Trigonometric "9* -* 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.

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.

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.

KEY CONCEPT

General Definitions of Trigonometric Functions

Let 0 be an angle in standard position, and let (x, y) be the point where the terminal side of 0 intersects the circle x2 + y2 = r2. The six trigonometric functions of 0 are defined as follows:

r rout mistook,

sin 0 =

cos 0 = — r

tan 0 = i , x * 0 x

esc 0 = —, y # 0 y

sec 0 = —, x + 0 x

cot 0 = ^ , y ¥= 0 y

; jj These functions are sometimes called circular functions.

(iEXAMPLE 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 0.

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 0 = - = 1 r 5

esc e = — = — y 3

cos 0 = - = - 4 tan 0 = i- = —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