用sinθ与cosθ表示sin6θ与cos6θ 能利用(cosθ+sinθi)^6=cos6θ+sin6θ 求出sin6
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用sinθ与cosθ表示sin6θ与cos6θ 能利用(cosθ+sinθi)^6=cos6θ+sin6θ 求出sin6θ=?cos6θ=?
![用sinθ与cosθ表示sin6θ与cos6θ 能利用(cosθ+sinθi)^6=cos6θ+sin6θ 求出sin6](/uploads/image/z/19264167-63-7.jpg?t=%E7%94%A8sin%CE%B8%E4%B8%8Ecos%CE%B8%E8%A1%A8%E7%A4%BAsin6%CE%B8%E4%B8%8Ecos6%CE%B8+%E8%83%BD%E5%88%A9%E7%94%A8%EF%BC%88cos%CE%B8%2Bsin%CE%B8i%29%5E6%3Dcos6%CE%B8%2Bsin6%CE%B8+%E6%B1%82%E5%87%BAsin6)
(cosθ+isinθ)^6
=cos^6θ+6icos^5θsinθ-15cos^4θsin^2θ-20icos^3θsin^3θ+15cos^2θsin^4θ+6icosθsin^5θ-sin^6θ
=(cos^6θ-15cos^4θsin^2θ+15cos^2θsin^4θ-sin^6θ)+i(6cos^5θsinθ-20cos^3θsin^3θ+6cosθsin^5θ)=cos6θ+sin6θ
∴cos6θ=cos^6θ-15cos^4θsin^2θ+15cos^2θsin^4θ-sin^6θ
sin6θ=6cos^5θsinθ-20cos^3θsin^3θ+6cosθsin^5θ
=cos^6θ+6icos^5θsinθ-15cos^4θsin^2θ-20icos^3θsin^3θ+15cos^2θsin^4θ+6icosθsin^5θ-sin^6θ
=(cos^6θ-15cos^4θsin^2θ+15cos^2θsin^4θ-sin^6θ)+i(6cos^5θsinθ-20cos^3θsin^3θ+6cosθsin^5θ)=cos6θ+sin6θ
∴cos6θ=cos^6θ-15cos^4θsin^2θ+15cos^2θsin^4θ-sin^6θ
sin6θ=6cos^5θsinθ-20cos^3θsin^3θ+6cosθsin^5θ
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