2. Copyright © 2007 Pearson Education, Inc. Slide 8-2 Chapter 8: Trigonometric Functions And Applications 8.1Angles and Their Measures 8.2Trigonometric Functions and Fundamental Identities 8.3Evaluating Trigonometric Functions 8.4Applications of Right Triangles 8.5The Circular Functions 8.6Graphs of the Sine and Cosine Functions 8.7Graphs of the Other Circular Functions 8.8Harmonic Motion

3. Copyright © 2007 Pearson Education, Inc. Slide 8-3 8.5The 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.

4. Copyright © 2007 Pearson Education, Inc. Slide 8-4 8.5 The Circular Functions Circular Functions

5. 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.

6. Copyright © 2007 Pearson Education, Inc. Slide 8-6 8.5Evaluating Circular Functions Example Evaluate and. Solution An angle of radians intersects the unit circle at the point (0, -1)

7. 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.

8. Copyright © 2007 Pearson Education, Inc. Slide 8-8 8.5Evaluating Circular Functions Example (a) Find the exact values of and. (b) Find the exact value of.

9. Copyright © 2007 Pearson Education, Inc. Slide 8-9 8.5Evaluating Circular Functions Solution (a) From the figure (b)The angle -5 p /3 radians is coterminal with an angle of p /3 radians. From the figure

10. 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.

11. 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.

12. 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

13. 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