1. 10-1 Sum and Difference
Formulas for Sine and
Cosine
Objectives:
1. Derive and apply sum and
difference formulas for sine and
cosine.

2. Investigation
 Evaluate each expression:
 cos (45° - 30°) =
 cos 45° - cos 30° =
 sin (60° - 45°) =
 sin 60° - sin 45° =
 What do you notice?

3. Deriving cos(α - β)
 By law of cosines:

4.  By the distance formula:
 Therefore:

5. cos(α + β)
 Remember:
 cos (-β) = cos β
 sin (- β) = - sin β
 So:

6. Deriving sin(α + β)
 Using the cofunction relationship
 Then:

7. sin(α - β)
 Replacing β with –β gives:

8. Formula Recap

9. Rewriting
 Rewrite each angle as a sum or
difference using special angles from
the unit circle:
 285°
 75°
 -15°
 -165°

10. Example 1:
 Find the exact value of sin 15°.

11. Example 2:
 Find the exact value of:

12. You Try!
 Find the exact value of each:
 cos 15°


13. Example 3:
 Show that

14. Example 4:
 Prove that

15. You Try!
 Prove that

16. Example 5:
 Suppose that and

17. You Try!
 Suppose that and