高一三角函数证明题
已知:sinθ=asinγ,tanθ=btanγ,其中θ为锐角,求证:cosθ=√[(a^2-1)/(b^2-1)]
人气:219 ℃ 时间:2020-02-03 10:41:33
解答
已知sinα=asinβ,tanα=btanβ,α为锐角,求证:(cosα)^2=(a^2-1)/(b^2-1).
(sinβ)^2=(sinα)^2/a^2,(cosβ)^2=1-(sinβ)^2=[a^2-(sinα)^2]/a^2.
(tanβ)^2=(sinα)^2/[a^2-(sinα)^2].
(tanα)^2=b^2*(tanβ)^2=b^2(sinα)^2/[a^2-(sinα)^2].
1/(cosα)^2=b^2/[a^2-(sinα)^2].
b^2(cosα)^2=a^2-1+(cosα)^2.
(cosα)^2=(a^2-1)/(b^2-1).
cosα=√[(a^2-1)/(b^2-1)]
此题(α=θ,β=γ)
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