DOUBLE-ANGLE AND HALF-ANGLE FORMULAS

1. DOUBLE-ANGLE AND HALF-ANGLE FORMULAS

2. If we want to know a formula for we could use the sum formula. we can trade these places This is called the double angle formula for sine since it tells you the sine of double 

3. Let's try the same thing for This is the double angle formula for cosine but by substiuting some identities we can express it in a couple other ways.

4. Double-angle Formula for Tangent

5. Summary ofDouble-Angle Formulas

6. We can also derive formulas for an angle divided by 2. Half-Angle Formulas As stated it is NOT both + and - but you must figure out where the terminal side of the angle is and put on the appropriate sign for that quadrant.

8. We could find sin 15° using the half angle formula. Since 15° is half of 30° we could use this formula if  = 30° 30° 30° 15° is in first quadrant and sine is positive there so we want the +

9. Let's draw a picture. 5 4   -3 Use triangle to find values.

10. If  is in quadrant II thenhalf  would be in quadrant I where sine is positive 5 4   -3 Use triangle to find cosine value.

11. Your Turn: Simplify an Expression • Simplify cot x cos x + sin x. • Click for answer.

12. Your Turn: Cosine Sum and Difference Identities • Find the exact value of cos 75°. • Click for answer.

13. Your Turn: Sine Sum and Difference Identities • Find the exact value of . • Click for answer.

14. Your Turn: Double-Angle Identities • If , find sin 2x given sin x < 0. • Click for answer.

15. Your Turn: Double-Angle Identities

16. Your Turn: Half-Angle Identities • Use a half-angle identity to find sin 22.5°. • Click for answer.

17. Verifying An Identity Using Double Angle Objective: 7-4 Double-Angle and Half-Angle Identities

18. Find using the double angle formulas. (no calculator) 1. sin 420° 2. 3. tan 240° 4. 5. cos 300° 6. tan 630°

19. Find the exact values of sin 2x, cos 2x, and tan 2x using the double angles formulas 1. 2.