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 of Double-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.
30°
30°
Since 15° is half of 30° we could use this formula if  = 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 then half  would be in quadrant I where sine is positive5 4   -3
Use triangle to find cosine value.

11. Your Turn: Simplify an Expression
Simplify cot x cos x + sin x.
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Page 189

12. Your Turn: Cosine Sum and Difference Identities
Find the exact value of cos 75°.
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Page 198

13. Your Turn: Sine Sum and Difference Identities
Find the exact value of
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14. Your Turn: Double-Angle Identities
If , find sin 2x given sin x < 0.
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15. Your Turn: Double-Angle Identities

16. Your Turn: Half-Angle Identities
Use a half-angle identity to find sin 22.5°.
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Page 223

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° tan 240° 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.