已知锐角α,β,γ满足cosα=tanβ;cosβ=tanγ;cosγ=tanα, 求sinα的值
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cosα=tanβ;cosβ=tanγ;两式相乘得cosαtanγ=tanβ×cosβ=sinβ
所以(sinβ)^2=(cosαtanγ)^2; (cosβ)^2=(tanγ)^2
则1=(cosβ)^2+(sinβ)^2=(tanγ)^2[1+(cosα)^2]
=[(sinγ)^2/(cosγ)^2][1+(cosα)^2]
=[(1-(cosγ)^2)/(cosγ)^2][1+(cosα)^2]
=[(1-(tanα)^2)/(tanα)^2][1+(cosα)^2]
=[(1-2(sinα)^2)/(sinα)^2)][2-(sinα)^2]
所以[(1-2(sinα)^2)][2-(sinα)^2]=(sinα)^2
(sinα)^4-3(sinα)^2+1=0
(sinα)^2=(3-√5)/2
sinα=(√5-1)/2
所以(sinβ)^2=(cosαtanγ)^2; (cosβ)^2=(tanγ)^2
则1=(cosβ)^2+(sinβ)^2=(tanγ)^2[1+(cosα)^2]
=[(sinγ)^2/(cosγ)^2][1+(cosα)^2]
=[(1-(cosγ)^2)/(cosγ)^2][1+(cosα)^2]
=[(1-(tanα)^2)/(tanα)^2][1+(cosα)^2]
=[(1-2(sinα)^2)/(sinα)^2)][2-(sinα)^2]
所以[(1-2(sinα)^2)][2-(sinα)^2]=(sinα)^2
(sinα)^4-3(sinα)^2+1=0
(sinα)^2=(3-√5)/2
sinα=(√5-1)/2
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