Double-and-Half-Anccccccccccccccgle-Formula.pptx

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1. DOUBLE AND HALF-ANGLE FORMULAS

2. Evaluate each for x = 60° sin 2x 2 sin x sin ½ x ½ sin x What do you notice? Investigation

3. If we let α = β in our angle addition formulas, we get: Double-Angle Formulas

4. Since we have two alternative formulas for cos 2 α : Alternative Forms

5. tan 2 α Again, letting α = β :

6. If and , find sin 2α , cos 2α , and tan 2 α . Example 1

7. If and , find sin 2α , cos 2α , and tan 2 α . Example 1

8. If and , find sin 2α , cos 2α , and tan 2 α . Example 1

9. Simplify: Example 2

10. You Try! Simplify:

11. Example 3: Derive a formula for sin 4x in terms of functions of x.

12. Half-Angle Formulas The sine and cosine formulas use the alternate forms for cos 2 α . Replacing α with gives:

13. sin x/2

14. cos x/2

15. Choosing Signs Choose + or – based on the quadrant where is located. Example: For (or cos 75°) choose + (since 75° is in the first quadrant)

16. Example 4: Fine the exact value of

17. tan x/2 Dividing sin x/2 by cos x/2 gives:

18. Alternate forms of tan x/2 Simplifying the radical gives:

19. Example 5: Prove the identity:

20. Example 6: Simplify: