10.3 Double and Half-Angle FormulasObjectives:

1. Derive and apply double-angle and half-angle formulas

Background/IntroTrig functions are used in science

and engineering to study light and sound waves.

Some sample sound waves:

InvestigationEvaluate each for x = 60°sin 2x2 sin x

sin ½ x½ sin x

What do you notice?

Double-Angle Formulas If we let α = β in our angle addition

formulas, we get:

Alternative Forms

Since we have two alternative formulas for cos 2α :

tan 2α

Again, letting α = β:

Example 1:

If and , find sin 2α, cos 2α, and tan 2α.

Example 2:

Simplify:

You Try!

Simplify:

Example 3:

Derive a formula for sin 4x in terms of functions of x.

Half-Angle FormulasThe sine and cosine formulas use the

alternate forms for cos 2α.Replacing α with gives:

sin x/2

cos x/2

Choosing Signs

Choose + or – based on the quadrant where is located.

Example: For (or cos 75°) choose +

(since 75° is in the first quadrant)

Example 4:

Fine the exact value of

tan x/2

Dividing sin x/2 by cos x/2 gives:

Alternate forms of tan x/2

Simplifying the radical gives:

Example 5:

Prove the identity:

Example 6:

Simplify: