DOUBLE-ANGLE AND HALF-ANGLE IDENTITIES

1. DOUBLE-ANGLE AND HALF-ANGLE IDENTITIES

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 x

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

4. Double-angle Formula for Tangent

5. Summary ofDouble-Angle Formulas

6. Let's draw a picture. 5 4 x x’ -3 Use triangle to find values.

7. We can also derive formulas for an angle divided by 2 (called the half angle formula). We’ll do this by using the double angle formula for cosine that we found. Let’s solve this for sin  Now let  = x/2 In this formula 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 can also derive a half angle formula for cosine in a similar manner. We’ll do this by using a different version of the double angle formula for cosine. Let’s solve this for cos  Now let  = x/2 In this formula 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.

9. Now to derive a half angle formula for tangent, let’s use the fact that we know that tangent is sine over cosine and use their half angle formulas.

10. Summary 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.

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

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