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Compute the exact value of 16·sin 20°·sin 40°·sin 60°·sin 80°. Give a…
Question
Compute the exact value of 16·sin 20°·sin 40°·sin 60°·sin 80°. Give a single integer.
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Step-by-step solution
Use the identity sin θ · sin(60° − θ) · sin(60° + θ) = (1/4) sin 3θ. With θ = 20°: sin 20°·sin 40°·sin 80° = sin 20°·sin(60° − 20°)·sin(60° + 20°) = (1/4) sin 60° = (1/4)(√3/2) = √3/8.
Then sin 20°·sin 40°·sin 60°·sin 80° = (√3/8)·sin 60° = (√3/8)(√3/2) = 3/16.
Multiplying by 16: 16·(3/16) = 3.
Final answer3
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