设tan(θ/2)=t,求证:sinθ=2t/(1+t^2),cosθ=(1-t^2)/(1+t^2),tanθ=2t/(1-t^2)
问题描述:
设tan(θ/2)=t,求证:sinθ=2t/(1+t^2),cosθ=(1-t^2)/(1+t^2),tanθ=2t/(1-t^2)
答
sinx=2sin(x/2)cos(x/2)
=2sin(x/2)cos(x/2)/(sin²(x/2)+cos²(x/2))
=2tan(x/2)/(tan²(x/2)+1)
=2t/(1+t²)
cosx=cos²(x/2)-sin²(x/2)
=(cos²(x/2)-sin²(x/2))/(sin²(x/2)+cos²(x/2))
=(1-tan²(x/2))/(tan²(x/2)+1)
=(1-t²)/(1+t²)
tanx=sinx/cosx
=2t/(1-t²)哪里来的x?我这里的x就是θθ键盘上不好输入用x代替oo