10-1 Sum and Difference Formulas for Sine and CosineObjectives:

1. Derive and apply sum and difference formulas for sine and cosine.

InvestigationEvaluate each expression:cos (45° - 30°) = cos 45° - cos 30° = sin (60° - 45°) = sin 60° - sin 45° = What do you notice?

Deriving cos(α - β)

By law of cosines:

By the distance formula:

Therefore:

cos(α + β)Remember: cos (-β) = cos βsin (- β) = - sin βSo:

Deriving sin(α + β)Using the cofunction relationship

Then:

sin(α - β)

Replacing β with –β gives:

Formula Recap

RewritingRewrite each angle as a sum or

difference using special angles from the unit circle:

285°75°-15°-165°

Example 1:

Find the exact value of sin 15°.

Example 2:Find the exact value of:

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

Example 3:

Show that

Example 4:

Prove that

You Try!

Prove that

Example 5:

Suppose that and

You Try!

Suppose that and