. , .

Trigonometric Functions on the Unit Circle 1

' • i mmm,. • n • - . I...,M i , 1 11 i

You found values of trigonometric functions for acute angles using ratios in right triangles. (Lesson 4-1)

NewVocabulary quadrantal angle reference angle unit circle circular function periodic function period

Find values of trigonometric functions for any angle.

| Find values of i trigonometric

functions using the unit circle.

A blood pressure of 120 over 80, measured in millimeters of mercury means that a person's blood pressure oscillates or cycles between 20 millimeters above and below a pressure of 100 millimeters of mercury for a given time t in seconds. A complete cycle of this oscillation takes about 1 second.

If the pressure exerted by the blood at time t = 0.25 second is 120 millimeters of mercury, then at time t = 1.25 seconds the pressure is also 120 millimeters of mercury.

1Trigonometric Functions of Any Angle i n Lesson 4-1, the definitions of the six trigonometric functions were restricted to positive acute angles. I n this lesson, these

definitions are extended to include any angle.

Concept Trigonometric Functions of Any Angle

Let 9 be any angle in standard position and point P{x, y) be a point on the terminal side of 9. Let r represent the nonzero distance from Pto the origin.

That is, let r= \jx2 + y2 ± 0. Then the trigonometric functions of 9 are as follows.

sin 9 =

cos9 = j

tan

csc0 = ^ , y ^ O

sec 9 = ^x^0

xj=0 cot 9

y

• x (°>

\y X

D(x,y)Vl_ y

Example Evaluate Trigonometric Functions Given a Point

Let (8, —6) be a point on the terminal side of an angle 0 in standard position. Find the exact values of the six trigonometric functions of 9.

Use the values of x and y to f i n d r.

Pythagorean Theorem

x = 8 and y = —6

Take the positive square root.

yjx2 + y1

V 8 2 + ( - 6 ) 2

VlOO or 10

y

(8, - 6 ) V

Use x = 8, y = —6, and r = 10 to write the six trigonometric ratios.

y - 6 „ „ 3 „ _ n x 8 4 t _ n V sin 9

8 4 1 0 ° r 5

8 4

tan 9 =

cot 9 =

or

• Guided Practice

The given point lies on the terminal side of an angle 9 i n standard posit ion. F ind the values of the six trigonometric functions of 6.

1A. (4 ,3 ) 1B. ( - 2 , - 1 )

fi§ 242 ] L e s s o n 4-3

Tip tel Angles There are "any quadrantal angles ^terminal with the s angles listed at the z measure of a quadrantal

i multiple of 90° or-J.

In Example 1, you found the trigonometric values of 0 without knowing the measure of 6. N o w we w i l l discuss methods for f inding these function values when only 9 is known. Consider trigonometric functions of quadrantal angles. When the terminal side of an angle 0 that is i n standard position lies on one of the coordinate axes, the angle is called a quadrantal angle.

...Concept Common Quadrantal Angles

y (0.1)'

0

y y

0 , ( ^ , . 0 (r, 0) x

0 X Vr.o) 0 X O X

( 0 , - r )

0 = 0° or 0 radians 0 = 90° or ^ radians 9 = 180° or IT radians 0 = 270° or ^-radians

You can f ind the values of the trigonometric functions of quadrantal angles by choosing a point on the terminal side of the angle and evaluating the function at that point. A n y point can be chosen. However, to simplify calculations, pick a point for which r equals 1.

valuate Trigonometric Functions of Quadrantal Angle! Find the exact value of each trigonometric funct ion, i f defined. I f not defined, wri te undefined.

a. s in ( - 1 8 0 ° )

The terminal side of —180° i n standard position lies on the negative x-axis. Choose a point P on the terminal side of the angle. A convenient point is (—1, 0) because r = 1.

sin ( -180°)

b. tan

^ o r 0

Sine function

y = 0 and r = 1

3ir The terminal side of — i n standard position lies on the negative y-axis. Choose a point P(0, —1) on the terminal side of the angle because r = 1.

tan

or undefined

Tangent function

y = —1 and x = 0

C. sec 4ir

The terminal side of 4TT in standard position lies on the positive x-axis. The point (1, 0) is convenient because r = 1.

sec 4TT

or 1

GuidedPractice

2A. cos 270°

Secant function

r= 1 and x= 1

2B. c s c ^

y

P ( - 1 , 0 )

-180° J *

P(1,0)

4TV

2C. c o t ( - 9 0 c

5 connectED.mcgraw-hill.com I 243

To f ind the values of the trigonometric functions of angles that are neither acute nor quadrantal, consider the three cases shown below in which a and b are positive real numbers. Compare the values of sine, cosine, and tangent of 9 and 9'.

StudyTip Reference Angles Notice that in some cases, the three trigonometric values of 9 and 0'(read theta prime) are the same. In other cases, they differ only in sign.

Quadrant III Quadrant IV

i-a, b)

a O

sin 9 = j

cos 9 =

tan 9 =

sin 9' •

cos 0' =

tan 9' tan 0':

(a, -b)

sin 0 = b r sin 0 ' =

cos 0 = a r cos 0 ' =

tan 0 = b a tan 0 ' =

This angle 9', called a reference angle, can be used to f ind the trigonometric values of any angle 9

| Concept Reference Angle Rules

If 0 is an angle in standard position, its reference angle 0' is the acute angle formed by the terminal side of 0 and the x-axis. The reference angle 0'for any angle 0,0° < 0 < 360° or 0 < 0 < 2ir, is defined as follows.

Quadrant II

y Quadrant

y Quadrant IV

y 0,

0'

9'=9 0 ' = 180° - 9

9' = TT - 0

0 - 1 8 0 °

0 - TV

0' = 360° - (

0' = 2-K - 0

To f ind a reference angle for angles outside the interval 0° < 9 < 360° or 0 < 9 < 27T, first f ind a corresponding coterminal angle i n this interval.

• j&u l j fe 3 Find Reference Angles Sketch each angle. Then find its reference angle.

a. 300 c b. - S L

The terminal side of 300° lies i n Quadrant IV. Therefore, its reference angle is 9' = 360° - 300° or 60°.

A coterminal angle is 27T — or . The

terminal side of lies i n Quadrant I I I , so

its reference angle is - j - — 7r or - j .

y 9 = 300°

, r [\ • V x

\ 0 ' = 6 O ° 0'

GuidedPractice 3A. 3B. •240° 3C. 390c

244 | L e s s o n 4-3 | Tr igonometric Functions on t h e Unit Circle

Because the tr igonometric values of an angle and its reference angle are equal or dif fer only i n s ign, y o u can use the f o l l o w i n g steps to f i n d the value of a t r igonometr ic funct ion of any angle 9.

KeyConcept Evaluating Trigonometric Functions of Any Angle

K i w i Find the reference angle 6'.

EflSflW Find the value of the trigonometric function for 9'.

PT7!flFl Using the quadrant in which the terminal side of 9 lies, determine the sign of the trigonometric function value of 9.

-Tip mg Trigonometric Values

the exact values of ?v 30°, 45°, 60°, and 90°, r e following pattern.

\orO

2

_ \_2 2

= V3 2

= £ o M

:s:tern exists for the --TCion, except the values

s /erse order.

The signs of the trigonometric functions i n each quadrant can be determined using the function definitions given on page 242. For example, because sin 9 = y, i t follows that sin 9 is negative

when y < 0, which occurs in Quadrants I I I and IV. Using this same logic, you can verify each of the signs for sin 9, cos 9, and tan 9 shown i n the diagram. Notice that these values depend only on x and y because r is always positive.

Because you know the exact trigonometric values of 30°, 45°, and 60° angles, you can f ind the exact trigonometric values of all angles for which these angles are reference angles. The table lists these values for 9 in both degrees and radians.

y

Quadrant II Quadrant 1 sin 9: + sin 9: +

cos 9: - cos 9: + tan 9: - tan 0: +

Quadrant III Quadrant IV sin 9: - sin 9: -cos 9: - cos 9: + tan 9: + tan 9: -

30° or ^ 6

45° o r £ 4

6 0 ° o r f

sin 9 1 V2 \/3 sin 9 2 2 2

cos 9 V3 >/2 1 cos 9 2 2 2

tan 9 V3 3 1 \

frt-iiiM;- ; Use Reference Angles to Find Trigonometric Values

Find the exact value of each expression,

a. cos 120°

Because the terminal side of 9 lies i n Quadrant I I , the reference angle 9' is 180° - 120° or 60°.

cos 120c -cos 60° 1

"2

In Quadrant I!, cos 0 is negative,

cos 60° = I

b. tan 21

Because the terminal side of 9 lies i n Quadrant I I I ,

V y

0 ' = 6 O ° / y - ^ 0 = 120°

0 X

the reference angle 9' is

7TT

7TT 7T or

tan tan

V3 3

6 6

In Quadrant III, tan 9 is positive.

t a n f = # 6 3

connectED.mcgraw-hill.com I 245

C. CSC 157V

15TT 77T A coterminal angle of 9 is — 2 7 T or — , which lies i n

Quadrant IV. So, the reference angle 9' is 2TT — — or —.

Because sine and cosecant are reciprocal functions and sin 9 is negative in Quadrant IV, it follows that esc 9 is also negative in Quadrant IV.

C S C I ^ L = _ C S C Z 4 4

sin

1

2

•VI

In Quadrant IV, esc 6 is negative,

esc 0 1 sin 0

CHECK You can check your answer by using a graphing calculator.

15TT •1.414 •

-\fl ~ -1 .414*/

p Guided Practice

Find the exact value of each expression.

4A. t a n ^ 4B. sin 5TT

((p

4 C sec(-135°)

If the value of one or more of the trigonometric functions and the quadrant in which the terminal side of 9 lies is known, the remaining function values can be found.

WatchOut! Rationalizing the Denominator Be sure to rationalize the denominator, if necessary.

Example 3 Use One Trigonometric Value to Find Others

5 Let tan 9 = — , where sin 9 < 0. F ind the exact values of the f ive remaining trigonometric

functions of 9.

To f i n d the other function values, you must f ind the coordinates of a point on the terminal side of 9. You know that tan 9 is positive and sin 9 is negative, so 9 must lie i n Quadrant I I I . This means that both x and y are negative.

Because tan 9 = ^ or use the point (—12, —5) to f ind r.

V * 2 + y2 Pythagorean Theorem

= V ( - 1 2 ) 2 + ( - 5 ) 2 x= - 1 2 and y = - 5

= V l 6 9 o r 13 Take the positive square root.

Use x = —12,y= —5, and r = 13 to write the five remaining trigonometric ratios.

12 0 x

( - 5 , - 1 2 )

sin 9 = - o r r 13

cos0 = - o r - ^ | -r 13

CSC sec

cot0 = | o r ^ -

x 12

GuidedPractice

Find the exact values of the f ive remaining trigonometric functions of 9.

5A. sec 9 = V 3 , tan 9 < 0 5B. sin 9 = y , cot 0 > 0

246 | L e s s o n 4-3 | T r igonometr ic Funct ions on the Unit Circle

Real-World Example Find Coordinates Given a Radius and an Angle

225°

WorldLmk s an international

or in which teams r 2 series of soccer depending on the size cexe of their robots. The I project is to advance rtelligence :cs research.

ROBOTICS As part of the range of mot ion category i n a h igh school robotics competit ion, a student programmed a 20-centimeter long robotic arm to pick u p an object at point C and rotate through an angle of exactly 225° i n order to release i t into a container at point D. F ind the posit ion of the object at point D, relative to the pivot point O.

With the pivot point at the origin and the angle through which the arm rotates in standard position, point C has coordinates (20,0). The reference angle 0'for 225° is 225° - 180° or 45°.

Let the position of point D have coordinates (x, y). The definitions of sine and cosine can then be used to f ind the values of x and y. The value of r, 20 centimeters, is the length of the robotic arm. Since D is i n Quadrant I I I , the sine and cosine of 225° are negative.

cos 6 = X r Cosine ratio

cos 225° = X

20 0 = 225° a n d r = 2 0

-cos 45° = X 20

cos 225° = -cos 45°

V2 _ 2

X 20

cos 45° = ^j-

- 1 0 V 2 = X Solve for x.

sin 9

sin 225°

- s i n 45°

2

1 0 V 2

y r y_

20

y_ 20

21)

Sine ratio

0 = 225° and r= 20

sin 225° •sin 45c

sin 45° =

Solve for y.

The exact coordinates of D are ( — 10V2 , —10>/2 ) . Since 1 0 \ / 2 is about 14.14, the object is about 14.14 centimeters to the left of the pivot point and about 14.14 centimeters below the pivot point.

p Guided Practice

6. CLOCKWORK A 3-inch-long minute hand on a clock shows a time of 45 minutes past the hour. What is the new position of the end of the minute hand relative to the pivot point at 10 minutes past the next hour?

Tip g Function The on of a point on the ne with a point on a circle

tie wrapping function, example, if w{t)

^ 3 s a point fon the number a point P(x, y) on the unit

Ten W(-K) = (-1,0) and = d,0).

2Trigonometric Functions on the Unit Circle a u n i t circle is a circle of radius 1 centered at the origin.

Notice thaton a unit circle, the radian measure of a central angle 0 = y or s, so the arc length intercepted by 9 corresponds

exactly to the angle's radian measure. This provides a way of mapping a real number input value for a trigonometric function to a real number output value.

Consider the real number line placed vertically tangent to the unit circle at (1, 0) as shown below. If this line were wrapped about the circle in both the positive (counterclockwise) and negative (clockwise) direction, each point t on the line w o u l d map to a unique point P(x, y) on the circle. Because r = 1, we can define the trigonometric ratios of angle t in terms of just x and y.

Positive Values of t Negative Values of t

y

(1,0)

KeyConcept Trigonometric Functions on the Unit Circle

Let fbe any real number on a number line and let P(x, y) be the point on t when the number line is wrapped onto the unit circle. Then the trigonometric functions of fare as follows.

sin t = y

csct=j,y±0

COS f = X

sec t=±,xj=0

tan t=j[,x± 0

c o U = ± y # 0

Therefore, the coordinates of P corresponding to the angle f can be written as P(cos t, sin t).

P (x, y) o r P(cos t, sin t)

Notice that the input value in each of the definitions above can be thought of as an angle measure or as a real number t. When defined as functions of the real number system using the unit circle, the trigonometric functions are often called circular functions.

Using reference angles or quadrantal angles, you should now be able to f ind the trigonometric function values for all integer multiples of 30°, or ^ radians, and 45°, or radians. These special values wrap to 16 special points on the unit circle, as shown below.

16-Point Unit Circle

y

StudyTip 16-Point Unit Circle You have already memorized these values in the first quadrant. The remaining values can be determined using the x-axis, y-axis, and origin symmetry of the unit circle along with the signs of xand y in each quadrant.

Using the (x, y) coordinates in the 16-point unit circle and the definitions i n the Key Concept Box at the top of the page, you can f i n d the values of the trigonometric functions for common angle measures. I t is helpful to memorize these exact function values so you can quickly perform calculations involving them.

fo-iiiijfo / Find Trignometric Values Using the Unit Circle

Find the exact value of each expression. I f undef ined, wr i te undefined.

a. s i n f

y corresponds to the point (x, y) = j o n the unit circle.

sin t = y

• TT V3 s i n — = r

Definition of sin t

y = ^ w h e n f = £ . 7 2 3

248 | L e s s o n 4-3 | Tr igonometr ic Funct ions on the Unit Circle

b. cos 135°

135° corresponds to the point (x, y)

cos t — x

V2

V 2 V 2 2 ' 2

on the unit circle.

cos 135°=

C. tan270 c

x = - 2 ^ L w h e n t= 135c

270° corresponds to the point (x, y) = (0, — 1) on the unit circle.

V

tan t = — Definition of tan t

tan 270° = ^=p x = 0 and y = - 1 , when t= 270°.

d. CSC

Therefore, tan 270° is undefined.

l lTT

7 7 corresponds to the point (x, y) = [ — - ™ ) c

Definition of esc f CSC t _ 1 y

_ J _ csc —— 6 1

2

- - 2

y = _ i w h e n r = : ^ . 2 6

Simplify.

w GuidedPractice 7A. cos 7B. s in l20 c 7C. cot210 c 7D. sec 7TT

iTyTip •» vs. Degrees While we ss: discuss one wrapping

"^conding to an angle J r :* 360°, this measure is S 3 : to a distance. On the rze one wrapping siC'- :s to both the angle xr-g 2-and the distance ~ L " : the circle.

As defined by wrapping the number line around the unit circle, the domain of both the sine and cosine functions is the set of all real numbers ( — 0 0 , 0 0 ) . Extending infinitely i n either direction, the number line can be wrapped mult iple times around the unit circle, mapping more than one f-value to the same point P(x, y) w i t h each wrapping, positive or negative.

- 1 < sin t < 1

(cos t, sin f) t

1 < cos t < 1

Because cos t = x, sin t = y, and one wrapping corresponds to a distance of 27T,

cos (t + 2w7r) = cos t and sin (t + 2WK) = sin t,

for any integer n and real number t.

StudyTip Periodic Functions The other three circular functions are also periodic. The periods of these functions will be discussed in Lesson 4-5.

The values for the sine and cosine function therefore lie i n the interval [—1,1] and repeat for every integer multiple of 2TT on the number line. Functions w i t h values that repeat at regular intervals are called periodic functions.

KeyConcept Periodic Functions

A function y = f(t) is periodic if there exists a positive real number c such that f(t + c) = f(t) for all values of t in the domain of

The smallest number cfor which f is periodic is called the period of f.

The sine and cosine functions are periodic, repeating values after 2TC, S O these functions have a period of 277. I t can be shown that the values of the tangent function repeat after a distance of 77 on the number line, so the tangent function has a period of TC and

tan t = tan (f + WK),

for any integer n and real number f, unless both tan t and tan (t + HTT) are undefined. You can use the periodic nature of the sine, cosine, and tangent functions to evaluate these functions.

Use the Periodic Nature of Circular Functions

Find the exact value of each expression.

IITC a. cos

cos

4

l lTT S ( ^ + 2TT Rewrite ~ £ as the sum of a number and 2ir.

4

COS 3u

V2 2

and + 27c map to the same point (x,

the unit circle.

cos f - xand x = - 2 ^ - w h e n t=^-. 2 4

b. sin |-

sin |-

2-K

3

3

) ) = s i n ( f : + 2 ( - l )7r )

s i n 4TT

2

Rewrite — ^ as the sum of a number and an integer multiple of 2ir.

~ - and ~ - 2 ( -1 )ir map to the same point (x, y)

the unit circle.

sin t = yand y = — ^ - w h e n t= 4p

on

c. tan

tan

19TT 6

197T tan (f+3*) 19it Rewrite - — as the sum of a number and an integer multiple of it.

6

tan ^ ~ and ~ + 3ir map to points on the unit circle with the same 6

^ tangent values.

-4= or ^ tan t = y~: x = ^ and y = 1 when f = V 3 ^

GuidedPractice

8A. s i n ^ 8B. cos ( - ^ ) 8 C t a n ^

Recall from Lesson 1-2 that a funct ion/ i s even i f for every x in the domain of/,/(—x) = / (x ) and odd if for every x in the domain of/, /(—x) = —f(x). You can use the unit circle to verify that the cosine function is even and that the sine and tangent functions are odd. That is,

cos (—f) = cos t sin (—t) = —sin t tan (—t) — —tan t.

250 [ L e s s o n 4-3 ] Tr igonometr ic Funct ions on the Unit Circle

Exercises = Step-by-Step Solutions begin on page R29.

en point lies on the terminal side of an angle 0 i n m i a r d position. F ind the values of the six trigonometric ffcrons of 9. (Example 1)

3.4)

- 4 , - 3 )

1 - 8 )

-5,15)

2. ( -6 ,6 )

4. (2,0)

6. ( 5 , - 3 )

8. ( - 1 , - 2 )

the exact value of each trigonometric funct ion, i f . If not def ined, wri te undefined. (Example 2)

10. tan 2TT

-180°) 12. esc 270°

| -270°) 14. sec 180°

I T 16. s e c ( - f

each angle. Then f i n d its reference angle. (Example

18. 210°

llTC 2 0 - 3

22. - 7 5 c

24.

the exact value of each expression.

7TT 4 - 26. 3

3TT 28. 4

390° 30.

llTT 32. 6

\e exact values of the f ive remaining trigonometric aions of 9. (Example 5)

s r . 9 = 2, where sin 9 > 0 and cos 9 > 0

ifc 9 = 2, where sin 9 > 0 and cos 9 < 0

1 where cos 9 > 0

where sin 9 < 0

I = V 3 , where sin 9 < 0 and cos 9 > 0

0 = 1 , where sin 9 < 0 and cos 9 < 0

0 = - 1 , where sin 9 < 0

5 12 13

where sin 9 > 0

(4?) CAROUSEL Zoe is on a carousel at the carnival. The diameter of the carousel is 80 feet. Find the position of her seat from the center of the carousel after a rotation of210 c rxampie t>)

42. COIN FUNNEL A coin is dropped into a funnel where it spins i n smaller circles unt i l i t drops into the bottom of the bank. The diameter of the first circle the coin makes is 24 centimeters. Before completing one fu l l circle, the coin travels 150° and falls over. What is the new position of the coin relative to the center of the funnel? (Example 6)

F ind the exact value of each expression. I f undef ined, wri te

undefined. (Examples 7 and 8)

43. sec 120°

l lTT 45. cos

47. esc 390°

49. esc 5400'

51. c o t /

44. sin 315°

46. t a n ( - f )

48. cot510 c

50. s e c ^

52. esc 177T

53. t a n - y 54. sec-^?

55. sin (• 5TT 56. c o s ^ 4

57. tan 1 4 7 T 58. cos 19TV\

6 }

59. RIDES Jae and Anya are on a ride at an amusement park. After the first several swings, the angle the ride makes w i t h the vertical is modeled by 9 = 22 cos irt, w i t h 9 measured in radians and t measured i n seconds. Determine the measure of the angle in radians for t = 0, 0.5,1,1.5, 2, and 2.5. <Example 8)

L r connectED.mcgraw-hill.com 1 251

Complete each trigonometric expression.

60. cos 60° = sin

62. sin 4 ? = cos

64. s i n ( - 4 5 c cos

61. tan - j - = sin 4

63. cos ^?- = sin

65. cos — = sin

66. ICE CREAM The monthly sales i n thousands of dollars for Fiona's Fine Ice Cream shop can be modeled by

-Kit - 4)

y = 713 + 59.6 sin — - — , where t = 1 represents

January t = 2 represents February, and so on. a. Estimate the sales for January, March, July, and

October. b. Describe w h y the ice cream shop's sales can be

represented by a trigonometric function.

Use the given values to evaluate the trigonometric functions.

67. cos(-0) = cos 9 = ?; sec 9 = ?

68. s in( -0 ) = p sin 9 = ?; esc 9 = ?

69. sec 0 = ||; cos 0 = ?; cos(-0) = ?

70. esc ; s i n # = ? ; s in( -0) = ?

71. GRAPHS Suppose the terminal side of an angle 9 i n standard position coincides w i t h the graph of y = 2x i n Quadrant I I I . Find the six trigonometric functions of 9.

Find the coordinates of P for each circle w i t h the given radius and angle measure.

72. iv 73. y

• IT \

\ " J x

y

2 i r \ ^ } V 0

5 ' X

74.

PUy)

75.

76. COMPARISON Suppose the terminal side of an angle 9l in standard position contains the point (7, —8), and the terminal side of a second angle 92 i n standard position contains the point (—7, 8). Compare the sines of 9X and 92-

77. TIDES The depth y i n meters of the tide on a beach varies as a sine function of x, the hour of the day. On a certain

f ( . - 4 ) + 8, where day, that function was y = 3 sin

x = 0 , 1 , 2, 24 corresponds to 12:00 midnight , 1:00 A . M , 2:00 A . M . , 12:00 midnight the next night.

a. What is the maximum depth, or high tide, that day?

b. A t what time(s) does the high tide occur?

78. ftgf MULTIPLE REPRESENTATIONS I n this problem, you w i l l investigate the period of the sine function.

a. TABULAR Copy and complete a table similar to the one below that includes all 16 angle measures from the unit circle.

1 • 7V 6

7T 4 3

2TT

sin 20

b. VERBAL After what values of 9 do sin 9, sin 29, and sin 49, repeat their range values? In other words, whai are the periods of these functions?

c. VERBAL Make a conjecture as to how the period of y = sin n9 is affected for different values of n.

H.O.T. Problems Use Higher-Order Thinking Skills

CHALLENGE For each statement, describe n.

a. cos n 0

b. esc [n • —1 is undefined.

REASONING Determine whether each statement is true or false. Explain your reasoning.

80. If cos 9 = 0.8, sec 9 - cos (-9) = 0.45.

81. Since tan (—t) = —tan t, the tangent of a negative angle is a negative number.

82. WRITING IN MATH Explain w h y the attendance at a year-round theme park could be modeled by a periodic function. What issues or events could occur over time to alter this periodic depiction?

REASONING Use the u n i t circle to ver i fy each relationship.

83. sin (—t) = —sin t

84. cos (—t) = cos t

85. tan (-t) = - t a n t

86. WRITING IN MATH Make a conjecture as to the periods of the secant, cosecant, and cotangent functions. Explain your reasoning.

252 I L e s s o n 4-3 I Tr igonometr ic Funct ions on t h e Unit Circle

Spiral Review

| each decimal degree measure i n D M S f o r m and each D M S measure i n decimal degree to the nearest thousandth. (Lesson 4-2)

~>.35c 88. 27.465c 89. 14° 5'20" 90. 173° 24'35"

EXERCISE A preprogrammed workout on a treadmill consists of intervals walking at various rates and angles of incline. A 1 % incline means 1 unit of vertical rise for every 100 units of horizontal run. (Lesson 4-1)

a. A t what angle, w i t h respect to the horizontal, is the treadmill bed when set at a 10% incline? Round to the nearest degree.

h. If the treadmill bed is 40 inches long, what is the vertical rise when set at an 8% incline?

angle of incline

vertical rise

h horizontal run >\

tate each logar i thm. (Lesson 3-3)

-'. 64 93. l o g 1 2 5 5 94. l o g 2 32 95. l o g 4 128

all possible rational zeros of each funct ion. Then determine w h i c h , i f any, are zeros.

4 x 2 + x + 2

x^ + X

= 2.v4 + 3 x J - 6x' llx

97. g(x) = x3 + 6x2 + Wx + 3

99. h(x) = 2x3 + 3x2 - 8x + 3

101. x(x) = 4 x 3 + * 2 + 8* + 2

•WH6ATI0N A global positioning system (GPS) uses satellites to allow a user to determine his or her position on Earth. The system depends on satellite signals that are reflected to and from a hand-held transmitter. The time that the signal takes to reflect is used to -aerermine the transmitter's position. Radio waves travel through air at a speed of 3^,792,458 meters per second. Thus, d(t) = 299,792,458^ relates the time t i n seconds to •fee distance traveled d(t) in meters. (Lesson 1-1)

3L Find the distance a radio wave w i l l travel i n 0.05, 0.2,1.4, and 5.9 seconds, fc- If a signal from a GPS satellite is received at a transmitter i n 0.08 second, how far from

the transmitter is the satellite?

Review for Standardized Tests

SAT/ACT In the figure, AB and AD are tangents to circle C. What is the value of m?

Note: Figure not drawn to scale.

Suppose 9 is an angle in standard position w i t h sin 9 > 0. I n which quadrant(s) could the terminal side of 9 lie?

A I only

B I and I I

C I and I I I

D I and IV

105. REVIEW Find the angular speed in radians per second of a point on a bicycle tire if it completes 2 revolutions in 3 seconds.

H

2

2TT 3

4 -

106. REVIEW Which angle has a tangent and cosine that are both negative?

A 110°

B 180°

C 210°

D 340°

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