10 3 double and half-angle formulas
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10.3 Double and Half-Angle FormulasObjectives:
1. Derive and apply double-angle and half-angle formulas
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Background/IntroTrig functions are used in science
and engineering to study light and sound waves.
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Some sample sound waves:
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InvestigationEvaluate each for x = 60°sin 2x2 sin x
sin ½ x½ sin x
What do you notice?
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Double-Angle Formulas If we let α = β in our angle addition
formulas, we get:
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Alternative Forms
Since we have two alternative formulas for cos 2α :
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tan 2α
Again, letting α = β:
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Example 1:
If and , find sin 2α, cos 2α, and tan 2α.
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Example 2:
Simplify:
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You Try!
Simplify:
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Example 3:
Derive a formula for sin 4x in terms of functions of x.
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Half-Angle FormulasThe sine and cosine formulas use the
alternate forms for cos 2α.Replacing α with gives:
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sin x/2
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cos x/2
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Choosing Signs
Choose + or – based on the quadrant where is located.
Example: For (or cos 75°) choose +
(since 75° is in the first quadrant)
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Example 4:
Fine the exact value of
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tan x/2
Dividing sin x/2 by cos x/2 gives:
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Alternate forms of tan x/2
Simplifying the radical gives:
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Example 5:
Prove the identity:
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Example 6:
Simplify: