Question

已知tanα=3，求三分之二sin²α+四分之一cos²α的值

需要详细的化简过程，谢谢！

Answer 1

2/3sin²α+1/4cos²α
=(2/3sin²α+1/4cos²α)/1
=(2/3sin²α+1/4cos²α)/(sin^2a+cos^2a)(上下同除cos^2a)
=(2/3tan^2a+1/4)/(tan^2a+1)
=(6+1/4)/10
=5/8

Answer 2

tanα=3 即sinα/cosα=3
平方得 sin²α=9cos²α
因为sin²α+cos²α=1 上市化为1-cos²α=9cos²α
cos²α=1/10
2/3sin²α+1/4cos²α
=2/3(1-cos²α)+1/4cos²α
=2/3-5/12cos²α
=2/3-1/24
=5/8

Answer 3

tan α=sin α/cos α=3 所以sin α=3 cos α 所以sinα^2=9 cosα^2 因为sinα^2+cosα^2=1 所以
cosα^2=1/10 所以2/3sinα^2+1/4cosα^2=5/8