The Unit CircleThe two historical perspectives of trigonometry incorporate different methods forintroducing the trigonometric functions. Our first introduction to these functionsis based on the unit circle.

Consider the unit circle given by

Unit circle

as shown in Figure 4.20.

FIGURE 4.20

Imagine that the real number line is wrapped around this circle, with positivenumbers corresponding to a counterclockwise wrapping and negative numberscorresponding to a clockwise wrapping, as shown in Figure 4.21.

FIGURE 4.21

As the real number line is wrapped around the unit circle, each real numbercorresponds to a point on the circle. For example, the real number 0

corresponds to the point Moreover, because the unit circle has a circum-ference of the real number also corresponds to the point

In general, each real number also corresponds to a central angle (instandard position) whose radian measure is With this interpretation of the arclength formula (with ) indicates that the real number is the lengthof the arc intercepted by the angle given in radians.�,

tr � 1s � r�t,t.

�t�1, 0�.2�2�,

�1, 0�.�x, y�t

(1, 0)

( , )x yt

t

t < 0

θ

y

x(1, 0)

( , )x yt

tt > 0

θ

y

x

x(1, 0)

(0, 1)

(0, 1)−

( 1, 0)−

y

x2 � y 2 � 1

294 Chapter 4 Trigonometry

What you should learn• Identify a unit circle and

describe its relationship to real numbers.

• Evaluate trigonometric functions using the unit circle.

• Use the domain and period to evaluate sine and cosinefunctions.

• Use a calculator to evaluatetrigonometric functions.

Why you should learn itTrigonometric functions are used to model the movement of an oscillating weight. Forinstance, in Exercise 57 on page300, the displacement from equilibrium of an oscillatingweight suspended by a spring is modeled as a function of time.

Trigonometric Functions: The Unit Circle

Richard Megna/Fundamental Photographs

4.2

Try demonstrating the wrappingfunction by using a spool and thread.Demonstrate the real number as thelength of the thread. Then wrap thethread around the spool to show thecorrespondence between and thepoint on the circle. For example, if

the point on the unit circle towhich it corresponds is �0, 1�.t � ��2,

�x, y�t

t

333202_0402.qxd 12/8/05 8:18 AM Page 294

The Trigonometric FunctionsFrom the preceding discussion, it follows that the coordinates and are twofunctions of the real variable You can use these coordinates to define the sixtrigonometric functions of

sine cosecant cosine secant tangent cotangent

These six functions are normally abbreviated sin, csc, cos, sec, tan, and cot,respectively.

In the definitions of the trigonometric functions, note that the tangent andsecant are not defined when For instance, because correspondsto it follows that and are undefined. Similarly,the cotangent and cosecant are not defined when For instance, because

corresponds to and csc 0 are undefined.In Figure 4.22, the unit circle has been divided into eight equal arcs, corre-

sponding to -values of

and

Similarly, in Figure 4.23, the unit circle has been divided into 12 equal arcs,corresponding to -values of

and

To verify the points on the unit circle in Figure 4.22, note that

also lies on the line So, substituting for in the equation of the unitcircle produces the following.

Because the point is in the first quadrant, and because you also

have You can use similar reasoning to verify the rest of the points in

Figure 4.22 and the points in Figure 4.23.Using the coordinates in Figures 4.22 and 4.23, you can easily evaluate

the trigonometric functions for common -values. This procedure is demonstratedin Examples 1 and 2. You should study and learn these exact function values for common -values because they will help you in later sections to performcalculations quickly and easily.

t

t�x, y�

y ��22

.

y � x,x ��22

x � ±�22

x2 �12

2x2 � 1x2 � x2 � 1

yxy � x.

��22

, �22 �

2�.0, �

6,

�

3,

�

2,

2�

3,

5�

6, �,

7�

6,

4�

3,

3�

2,

5�

3,

11�

6,

t

2�.0, �

4,

�

2,

3�

4, �,

5�

4,

3�

2,

7�

4,

t

�x, y� � �1, 0�, cot 0t � 0y � 0.

sec���2�tan���2��x, y� � �0, 1�,t � ��2x � 0.

t.t.

yx

Section 4.2 Trigonometric Functions: The Unit Circle 295

Definitions of Trigonometric FunctionsLet be a real number and let be the point on the unit circle correspon-ding to

y � 0cot t �x

y,x � 0sec t �

1

x,y � 0csc t �

1

y,

x � 0tan t �y

x,cos t � xsin t � y

t.�x, y�t

x(1, 0)

(0, 1)

(0, 1)−

( 1, 0)−

2 2

2 22 2

2 22 2

2 22 2

2 2, −

, ,

, −( )

( )

( )

( )

−

−

y

FIGURE 4.22

x

y

(1, 0)

(0, 1)

(0, 1)−

( 1, 0)−

3

3

3

3

3

3

3

3

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

, −

, −

,

, −

,

,,

, −

( )

( )

( )

( )

( )( )

( )

( )−

−

−

−

FIGURE 4.23

Note in the definition at theright that the functions in thesecond row are the reciprocalsof the corresponding functionsin the first row.

333202_0402.qxd 12/7/05 11:02 AM Page 295

Evaluating Trigonometric Functions

Evaluate the six trigonometric functions at each real number.

a. b. c. d.

SolutionFor each -value, begin by finding the corresponding point on the unitcircle. Then use the definitions of trigonometric functions listed on page 295.

a. corresponds to the point

b. corresponds to the point

c. corresponds to the point

is undefined.

is undefined.

d. corresponds to the point

is undefined.

is undefined.

Now try Exercise 23.

cot � �x

ytan � �

y

x�

0

�1� 0

sec � �1

x�

1

�1� �1cos � � x � �1

csc � �1

ysin � � y � 0

�x, y� � ��1, 0�.t � �

cot 0 �x

ytan 0 �

y

x�

0

1� 0

sec 0 �1

x�

1

1� 1cos 0 � x � 1

csc 0 �1

ysin 0 � y � 0

�x, y� � �1, 0�.t � 0

cot 5�

4�

x

y�

��2�2

��2�2� 1tan

5�

4�

y

x�

��2�2

��2�2� 1

sec 5�

4�

1

x� �

2�2

� ��2cos 5�

4� x � �

�2

2

csc 5�

4�

1

y� �

2�2

� ��2sin 5�

4� y � �

�2

2

�x, y� � ���2

2, �

�2

2 �.t �5�

4

cot �

6�

x

y�

�3�2

1�2� �3tan

�

6�

y

x�

1�2�3�2

�1�3

��3

3

sec �

6�

1

x�

2�3

�2�3

3cos

�

6� x �

�3

2

csc �

6�

1

y�

1

1�2� 2sin

�

6� y �

1

2

�x, y� � ��3

2,

1

2�.t ��

6

�x, y�t

t � �t � 0t �5�

4t �

�

6

296 Chapter 4 Trigonometry

Example 1

Additional Example

Evaluate the six trigonometric functions

at

Solution

Moving counterclockwise around theunit circle one and a quarter revolutions,you find that corresponds tothe point

is undefined.

is undefined.

cot 5�

2�

xy

� 0

tan 5�

2�

yx

sec 5�

2�

1x

cos 5�

2� x � 0

csc 5�

2�

1y

� 1sin 5�

2� y � 1

�x, y� � �0, 1�.t � 5��2

t �5�

2.

333202_0402.qxd 12/7/05 11:02 AM Page 296

Evaluating Trigonometric Functions

Evaluate the six trigonometric functions at

SolutionMoving clockwise around the unit circle, it follows that correspondsto the point

Now try Exercise 25.

Domain and Period of Sine and CosineThe domain of the sine and cosine functions is the set of all real numbers. Todetermine the range of these two functions, consider the unit circle shown inFigure 4.24. Because it follows that and Moreover,because is on the unit circle, you know that and So, the values of sine and cosine also range between and 1.

and

Adding to each value of in the interval completes a secondrevolution around the unit circle, as shown in Figure 4.25. The values of

and correspond to those of and Similarresults can be obtained for repeated revolutions (positive or negative) on the unitcircle. This leads to the general result

and

for any integer and real number Functions that behave in such a repetitive (orcyclic) manner are called periodic.

t.n

cos�t � 2� n� � cos t

sin�t � 2�n� � sin t

cos t.sin tcos�t � 2��sin�t � 2��

�0, 2��t2�

�1

�1

≤

≤

x

cos t

≤

≤

1

1

�1

�1

≤

≤

y

sin t

≤

≤

1

1

�1�1 ≤ x ≤ 1.�1 ≤ y ≤ 1�x, y�

cos t � x.sin t � yr � 1,

cot���

3� �1�2

��3�2� �

1�3

� ��3

3tan��

�

3� ���3�2

1�2� ��3

sec���

3� � 2cos���

3� �1

2

csc���

3� � �2�3

� �2�3

3sin��

�

3� � ��3

2

�x, y� � �1�2, ��3�2�.t � ���3

t � ��

3.

Section 4.2 Trigonometric Functions: The Unit Circle 297

Example 2

Definition of Periodic FunctionA function is periodic if there exists a positive real number such that

for all in the domain of The smallest number for which is periodic iscalled the period of f.

fcf.t

f�t � c� � f �t�

cf

With your graphing utility inradian and parametric modes,enter the equations

X1T = cos T and Y1T = sin T

and use the following settings.

Tmin = 0, Tmax = 6.3,Tstep = 0.1Xmin = -1.5, Xmax = 1.5,Xscl = 1Ymin = -1, Ymax = 1,Yscl = 1

1. Graph the entered equationsand describe the graph.

2. Use the trace feature tomove the cursor around thegraph. What do the -valuesrepresent? What do the

and values represent?

3. What are the least andgreatest values of and y?x

y-x-

t

Exploration

−1 ≤ x ≤ 1

−1 ≤ y ≤ 1(−1, 0) (1, 0)

(0, −1)

(0, 1)

y

x

FIGURE 4.24

x

y

t =

t =

,

,

+ 2

+ 2

t = , 3 , ...

t =

t =

2

4

4

4

4

4

2

4

2,

,

,+ 2

+ 2

+ 4 , ...

, ...

t = 0, 2 , ...

π

π

π

π

π

π

π

π

π

π π

π

π

π

π

π

π

, ...

, ...

5

3

5

3

t = 4 4 4, ,+ 2 + 4 , ...π π ππ π7 7 7

t = 2 2 2, ,+ 2 + 4 , ...π π ππ π3 3 3

FIGURE 4.25

333202_0402.qxd 12/7/05 11:02 AM Page 297

298 Chapter 4 Trigonometry

Recall from Section 1.5 that a function is even if and is oddif

Using the Period to Evaluate the Sine and Cosine

a. Because you have

b. Because you have

c. For because the sine function is odd.

Now try Exercise 31.

Evaluating Trigonometric Functions with a CalculatorWhen evaluating a trigonometric function with a calculator, you need to set thecalculator to the desired mode of measurement (degree or radian).

Most calculators do not have keys for the cosecant, secant, and cotangentfunctions. To evaluate these functions, you can use the key with their respec-tive reciprocal functions sine, cosine, and tangent. For example, to evaluate

use the fact that

and enter the following keystroke sequence in radian mode.

8 Display 2.6131259

Using a Calculator

Function Mode Calculator Keystrokes Display

a. Radian 2 3 0.8660254

b. Radian 1.5 0.0709148

Now try Exercise 45.

cot 1.5

sin 2�

3

csc �

8�

1

sin���8�

csc���8�,

sin��t� � �45

sin t �45

,

cos ��7�

2 � � cos��4� ��

2� � cos �

2� 0.

�7�

2� �4� �

�

2,

sin 13�

6� sin�2� �

�

6� � sin �

6�

1

2.

13�

6� 2� �

�

6,

f ��t� � �f �t�.f ��t� � f �t�,f

SIN � � � x �1 ENTER

SIN ENTER

�

� � �

x �1

Even and Odd Trigonometric FunctionsThe cosine and secant functions are even.

The sine, cosecant, tangent, and cotangent functions are odd.

cot��t� � �cot ttan��t� � �tan t

csc��t� � �csc tsin��t� � �sin t

sec��t� � sec tcos��t� � cos t

Example 3

From the definition of periodicfunction, it follows that the sineand cosine functions are peri-odic and have a period of The other four trigonometricfunctions are also periodic, andwill be discussed further inSection 4.6.

2�.

Students may have difficulty evaluatingcosecant, secant, and/or cotangentfunctions using a calculator.Try havingthem rewrite the expression in terms ofsine, cosine, or tangent before evaluating.

Example: csc 1.3 �1

sin 1.3 1.0378

When evaluating trigonometricfunctions with a calculator,remember to enclose all fractionalangle measures in parentheses.For instance, if you want to evaluate for youshould enter

6 .

These keystrokes yield the correctvalue of 0.5. Note that some calcu-lators automatically place a leftparenthesis after trigonometricfunctions. Check the user’s guidefor your calculator for specific keystrokes on how to evaluatetrigonometric functions.

� � ��6,sin �

Techno logy

SIN � � � ENTER

TAN �� x �1 ENTER

Example 4

333202_0402.qxd 12/7/05 11:02 AM Page 298

Section 4.2 Trigonometric Functions: The Unit Circle 299

Exercises 4.2

In Exercises 1– 4, determine the exact values of the sixtrigonometric functions of the angle

1. 2.

3. 4.

In Exercises 5–12, find the point on the unit circle thatcorresponds to the real number

5. 6.

7. 8.

9. 10.

11. 12.

In Exercises 13–22, evaluate (if possible) the sine, cosine,and tangent of the real number.

13. 14.

15. 16.

17. 18.

19. 20.

21. 22.

In Exercises 23–28, evaluate (if possible) the six trigono-metric functions of the real number.

23. 24.

25. 26.

27. 28.

In Exercises 29–36, evaluate the trigonometric functionusing its period as an aid.

29. 30.

31. 32.

33. 34.

35. 36.

In Exercises 37– 42, use the value of the trigonometricfunction to evaluate the indicated functions.

37. 38.

(a) (a)

(b) (b)

39. 40.

(a) (a)

(b) (b)

41. 42.

(a) (a)

(b) (b) cos�t � ��sin�t � ��cos�� � t�sin�� � t�

cos t �45sin t �

45

sec��t�sec��t�cos��t�cos t

cos t � �34cos��t� � �

15

csc tcsc��t�sin tsin��t�

sin��t� �38sin t �

13

cos��8�

3 �sin��9�

4 �

sin 19�

6cos��

15�

2 �

sin 9�

4cos

8�

3

cos 5�sin 5�

t �7�

4t �

4�

3

t �3�

2t � �

�

2

t �5�

6t �

3�

4

t � �2�t � �3�

2

t �5�

3t �

11�

6

t � �4�

3t � �

7�

4

t � ��

4t � �

�

6

t ��

3t �

�

4

t � �t �3�

2

t �5�

3t �

4�

3

t �5�

4t �

7�

6

t ��

3t �

�

4

t.x, y�

x

45

35

, −( (−

θ

y

x

12 513 13

, −( (

θ

y

x

12 513 13

, ( (

θ

y

x

8 1517 17( (− ,

θ

y

�.

VOCABULARY CHECK: Fill in the blanks.

1. Each real number corresponds to a point on the ________ ________.

2. A function is ________ if there exists a positive real number such that for all in the domain of

3. The smallest number for which a function is periodic is called the ________ of

4. A function is ________ if and ________ if

PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com.

f ��t� � f �t�.f ��t� � �f �t�f

f.fc

f.tf �t � c� � f �t�cf

�x, y�t

333202_0402.qxd 12/7/05 11:02 AM Page 299

In Exercises 43–52, use a calculator to evaluate the trigono-metric function. Round your answer to four decimal places.(Be sure the calculator is set in the correct angle mode.)

43. 44.

45. 46.

47. 48.

49. 50.

51. 52.

Estimation In Exercises 53 and 54, use the figure and astraightedge to approximate the value of each trigono-metric function. To print an enlarged copy of the graph, goto the website www.mathgraphs.com.

53. (a) sin 5 (b) cos 2 54. (a) sin 0.75 (b) cos 2.5

FIGURE FOR 53–56

Estimation In Exercises 55 and 56, use the figure and astraightedge to approximate the solution of each equa-tion, where To print an enlarged copy of thegraph, go to the website www.mathgraphs.com.

55. (a) (b)

56. (a) (b)

58. Harmonic Motion The displacement from equilibriumof an oscillating weight suspended by a spring is given by

where is the displacement (in feet) and isthe time (in seconds). Find the displacement when (a)

(b) and (c)

Synthesis

True or False? In Exercises 59 and 60, determine whetherthe statement is true or false. Justify your answer.

59. Because it can be said that the sine of anegative angle is a negative number.

60.

61. Exploration Let and be points on the unitcircle corresponding to and respectively.

(a) Identify the symmetry of the points and

(b) Make a conjecture about any relationship betweenand

(c) Make a conjecture about any relationship betweenand

62. Use the unit circle to verify that the cosine and secantfunctions are even and that the sine, cosecant, tangent, andcotangent functions are odd.

Skills Review

In Exercises 63– 66, find the inverse function of the one-to-one function

63. 64.

65. 66.

In Exercises 67–70, sketch the graph of the rational func-tion by hand. Show all asymptotes.

67. 68.

69. 70. f �x� �x3 � 6x2 � x � 1

2x2 � 5x � 8f �x� �

x2 � 3x � 102x2 � 8

f �x� �5x

x2 � x � 6f �x� �

2xx � 3

f �x� �x � 2x � 4

x ≥ 2f �x� � �x2 � 4,

f �x� �14 x3 � 1f �x� �

12�3x � 2�

f.f �1

cos�� � t1�.cos t1

sin�� � t1�.sin t1

�x2, y2�.�x1, y1�t � � � t1,t � t1

�x2, y2��x1, y1�

tan a � tan�a � 6��

sin��t� � �sin t,

t �12.t �

14,t � 0,

tyy�t� �14 cos 6t,

cos t � 0.75sin t � �0.75

cos t � �0.25sin t � 0.25

0 ≤ t < 2�.

0.25

0.50

0.75

1.001.251.501.75

2.002.25

2.50

2.75

3.00

3.25

3.50

3.75

4.004.25

4.50 4.75 5.005.25

5.50

5.75

6.00

6.25

0.2 0.4 0.6 0.8 1.2

0.2

0.4

0.6

0.8

−0.8

−0.6

−0.8

−0.4

−0.4

−0.2−0.6 −0.2

sin��0.9�sec 22.8

sec 1.8csc 0.8

cos��2.5�cos��1.7�cot 1csc 1.3

tan �

3sin

�

4

300 Chapter 4 Trigonometry

57. Harmonic Motion The displacement fromequilibrium of an oscillating weight suspended by aspring and subject to the damping effect of friction isgiven by where is the displace-ment (in feet) and is the time (in seconds).t

yy �t� �14e�t cos 6t

Model It

Model It (cont inued)

(a) Complete the table.

(b) Use the table feature of a graphing utility toapproximate the time when the weight reachesequilibrium.

(c) What appears to happen to the displacement as increases?

t

t 0 1

y

34

12

14

333202_0402.qxd 12/7/05 11:02 AM Page 300