1. Chapter 5 Analytic Trigonometry 1

2. 5.5 Multiple Angle Formulas Objective:  Rewrite and evaluate trigonometric functions using:  multiple-angle formulas  power-reducing formulas  half-angle formulas  product-to-sum and sum-to-product formulas 2

3. Double Angles  Use “sum” formulas to determine the double-angle formulas for sine, cosine, and tangent.  sin 2u = sin (u + u)  cos 2u = cos (u + u)  tan 2u = tan (u + u) 3

4. Double-Angle Formulas 4

5. Note:  Double-angle formulas are not restricted to the angles 2u and u.  Other double combinations, such as 4u and 2u or 6u and 3u, are also valid.  Two examples:  sin 4u = 2 sin 2u cos 2u  cos 6u = cos 2 3u – sin 2 3u 5

6. Example 1  Solve 2 cos x + sin 2x = 0. 6

7. Example 2  Use your graphing calculator to find the approximate solution(s) of the equation in the interval [0, 2π). If possible, find the exact solution(s) algebraically. 7

8. Example 3  Use the following to find sin 2θ, cos 2θ, and tan 2θ. 8

9. Example 4  Derive the triple-angle formula for sin 3x. Hint: sin 3x = sin (2x + x). 9

10. Power-Reducing Formulas 10

11. Example 5  Rewrite sin 4 x as a sum of first powers of the cosines of multiple angles. Hint: sin 4 x = (sin 2 x) 2 11

12. Alternate Forms  Start with the power-reducing formula for sine.  Replace u with u/2 to find the half-angle formula for sine.  Similarly, we can derive the half-angle formulas for cosine and tangent. 12

13. Half-Angle Formulas  How do we know whether to use + or – for sine and cosine? 13

14. Example 6  Find the exact value of sin 105 °. Hint: 105 ° is ½ of ______? 14

15. Example 7  Find all solutions in the interval [0, 2π). 15

16. Product-to-Sum Formulas – 1 16  We can verify these using sum and difference formulas – but we won’t.

17. Product-to-Sum Formulas – 2 17

18. Example 8  Rewrite the product as a sum or difference. cos 5x sin 4x 18

19. Sum-to-Product Formulas – 1 19

20. Product-to-Sum Formulas – 2 20

21. Example 9  Find the exact value of cos 195 ° + cos 105 °. 21

22. Example 10  Find all solutions in the interval [0, 2π). sin 5x + sin 3x = 0 22

23. Example 11  Verify the identity. 23

24. Homework 5.5  Worksheet 5.5 24