Copyright © 2007 Pearson Education, Inc. Slide 8-2

Chapter 8: Trigonometric Functions And Applications

8.1 Angles and Their Measures

8.2 Trigonometric Functions and Fundamental Identities

8.3 Evaluating Trigonometric Functions

8.4 Applications of Right Triangles

8.5 The Circular Functions

8.6 Graphs of the Sine and Cosine Functions

8.7 Graphs of the Other Circular Functions

8.8 Harmonic Motion

Copyright © 2007 Pearson Education, Inc. Slide 8-3

8.5 The Circular Functions

The trigonometric functions, when defined for all real values, are referred to as the circular functions. To define the circular functions for any real number s, use the unit circle, the circle with center at the origin and radius one unit. Start at the point (1,0) and measure an arc of length s, counterclockwise if s > 0 and clockwise if s < 0.

Copyright © 2007 Pearson Education, Inc. Slide 8-4

8.5 The Circular Functions

Circular Functions

sin cos tan ( 0)

1 1csc ( 0) sec ( 0) cot ( 0)

ys y s x s x

xx

s y s x s yy x y

Copyright © 2007 Pearson Education, Inc. Slide 8-5

8.5 The Circular Functions

Evaluating a Circular Function

Circular function values of real numbers are obtained in the same manner as trigonometric function values of angles measured in radians. This applies to both methods of finding exact values (such as reference angle analysis) and calculator approximations. Calculators must be in radian mode when finding circular function values.

Copyright © 2007 Pearson Education, Inc. Slide 8-6

8.5 Evaluating Circular Functions

Example Evaluate and .

Solution An angle of radians intersects the

unit circle at the point (0, -1)

3π 3πsin , cos2 2

3πtan2

3π2

3πsin so sin 123πcos so cos 023πtan so tan is undefined.2

s y

s x

ysx

Copyright © 2007 Pearson Education, Inc. Slide 8-7

8.5 Special Angles and The Circular Functions

The special angles and their corresponding points on the unit circle are summarized in the figure.

Copyright © 2007 Pearson Education, Inc. Slide 8-8

8.5 Evaluating Circular Functions

Example

(a) Find the exact values of and .

(b) Find the exact value of .

7πcos4

7πsin4

5πtan

3

Copyright © 2007 Pearson Education, Inc. Slide 8-9

8.5 Evaluating Circular Functions

Solution (a) From the figure

(b) The angle -5/3 radians is coterminal with an angle of /3 radians. From the figure

7π 2 7π 2cos , sin .4 2 4 2

35π π 2tan tan 3

13 32

Copyright © 2007 Pearson Education, Inc. Slide 8-10

8.5 Trigonometric Functions and the Unit Circle

•The figure relates trigonometric functions, triangles and the unit circle.

•Horizontal segments to the left of the origin and vertical segments below the x-axis represent negative values.

Copyright © 2007 Pearson Education, Inc. Slide 8-11

8.5 Applications of Circular Functions

The phase F of the moon is given by

where t is called the phase angle. F(t) gives the fraction of the moon’s face illuminated by the sun.

1( ) (1 cos )2

F t t

Copyright © 2007 Pearson Education, Inc. Slide 8-12

8.5 Modeling the Phases of the Moon

Example Evaluate and interpret.

(a) (b) (c) (d)

Solution

(a) new moon

(0)Fπ2

F

3π2

F

(π)F

1 1(0) (1 cos0) (1 1) 02 2

F

Copyright © 2007 Pearson Education, Inc. Slide 8-13

8.5 Modeling the Phases of the Moon

Solution

(b) first quarter

(c) full moon

(d) last quarter

1 1(π) (1 cos π) [1 ( 1)] 12 2

F

π

2

π 1 1 1(1 cos ) [1 0]2 2 2 2

F

3π

2

3π 1 1 1(1 cos ) [1 0]2 2 2 2

F