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Prove-sin5-16sin-5-20sin-3-5sin-Hence-show-that-sin-6-is-an-irrational-number-

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Question Number 106726 by ZiYangLee last updated on 06/Aug/20
Prove sin5θ=16sin^5 θ−20sin^3 θ+5sinθ Hence, show that sin 6° is an irrational number.
Provesin5θ=16sin5θ−20sin3θ+5sinθHence,showthatsin6°isanirrationalnumber.
Answered by Ar Brandon last updated on 06/Aug/20
sin5θ=Im(cos5θ+isin5θ)=Im(cosθ+isinθ)^5 =Im(C^5 +5iC^4 S−10C^3 S^2 −10iC^2 S^3 +5CS^4 +iS^5 ) =5C^4 S−10C^2 S^3 +S^5 =5(1−S^2 )^2 S−10(1−S^2 )S^3 +S^5 =5(1−2S^2 +S^4 )S−10(S^3 −S^5 )+S^5 =16S^5 −20S^3 +5S ⇒sin5θ=16sin^5 θ−20sin^3 θ+5sinθ
sin5θ=Im(cos5θ+isin5θ)=Im(cosθ+isinθ)5=Im(C5+5iC4S−10C3S2−10iC2S3+5CS4+iS5)=5C4S−10C2S3+S5=5(1−S2)2S−10(1−S2)S3+S5=5(1−2S2+S4)S−10(S3−S5)+S5=16S5−20S3+5S⇒sin5θ=16sin5θ−20sin3θ+5sinθ

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