1. Copyright © 2011 Pearson, Inc. 4.3 Trigonometry Extended: The Circular Functions

2. Copyright © 2011 Pearson, Inc. Slide 4.3 - 2 What you’ll learn about Trigonometric Functions of Any Angle Trigonometric Functions of Real Numbers Periodic Functions The 16-point unit circle … and why Extending trigonometric functions beyond triangle ratios opens up a new world of applications.

3. Copyright © 2011 Pearson, Inc. Slide 4.3 - 3 Initial Side, Terminal Side

4. Copyright © 2011 Pearson, Inc. Slide 4.3 - 4 Positive Angle, Negative Angle

5. Copyright © 2011 Pearson, Inc. Slide 4.3 - 5 Coterminal Angles Two angles in an extended angle-measurement system can have the same initial side and the same terminal side, yet have different measures. Such angles are called coterminal angles.

6. Copyright © 2011 Pearson, Inc. Slide 4.3 - 6 Example Finding Coterminal Angles

7. Copyright © 2011 Pearson, Inc. Slide 4.3 - 7 Example Finding Coterminal Angles

8. Copyright © 2011 Pearson, Inc. Slide 4.3 - 8 Example Finding Coterminal Angles

9. Copyright © 2011 Pearson, Inc. Slide 4.3 - 9 Example Finding Coterminal Angles

10. Copyright © 2011 Pearson, Inc. Slide 4.3 - 10 Example Evaluating Trig Functions Determined by a Point in QI

11. Copyright © 2011 Pearson, Inc. Slide 4.3 - 11 Example Evaluating Trig Functions Determined by a Point in QI

12. Copyright © 2011 Pearson, Inc. Slide 4.3 - 12 Trigonometric Functions of any Angle

13. Copyright © 2011 Pearson, Inc. Slide 4.3 - 13 Evaluating Trig Functions of a Nonquadrantal Angle θ 1. Draw the angle θ in standard position, being careful to place the terminal side in the correct quadrant. 2. Without declaring a scale on either axis, label a point P (other than the origin) on the terminal side of θ. 3. Draw a perpendicular segment from P to the x-axis, determining the reference triangle. If this triangle is one of the triangles whose ratios you know, label the sides accordingly. If it is not, then you will need to use your calculator.

14. Copyright © 2011 Pearson, Inc. Slide 4.3 - 14 Evaluating Trig Functions of a Nonquadrantal Angle θ 4.Use the sides of the triangle to determine the coordinates of point P, making them positive or negative according to the signs of x and y in that particular quadrant. 5.Use the coordinates of point P and the definitions to determine the six trig functions.

15. Copyright © 2011 Pearson, Inc. Slide 4.3 - 15 Example Evaluating More Trig Functions

16. Copyright © 2011 Pearson, Inc. Slide 4.3 - 16 Example Evaluating More Trig Functions

17. Copyright © 2011 Pearson, Inc. Slide 4.3 - 17 Example Using one Trig Ration to Find the Others

18. Copyright © 2011 Pearson, Inc. Slide 4.3 - 18 Example Using one Trig Ration to Find the Others

19. Copyright © 2011 Pearson, Inc. Slide 4.3 - 19 Example Using one Trig Ration to Find the Others

20. Copyright © 2011 Pearson, Inc. Slide 4.3 - 20 Unit Circle The unit circle is a circle of radius 1 centered at the origin.

21. Copyright © 2011 Pearson, Inc. Slide 4.3 - 21 Trigonometric Functions of Real Numbers

22. Copyright © 2011 Pearson, Inc. Slide 4.3 - 22 Periodic Function

23. Copyright © 2011 Pearson, Inc. Slide 4.3 - 23 The 16-Point Unit Circle

24. Copyright © 2011 Pearson, Inc. Slide 4.3 - 24 Quick Review

25. Copyright © 2011 Pearson, Inc. Slide 4.3 - 25 Quick Review Solutions