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: