已知sin(x+π/4)=4/5,cos(x+π/4)=-3/5,求sinx的值

问题描述:

已知sin(x+π/4)=4/5,cos(x+π/4)=-3/5,求sinx的值

由已知可得sin(x+π/4)+cos(x+π/4)=4/5+-3/5=1/5,则由辅助角公式可得
√2sin(x+π/4+π/4)=1/5 即√2sin(x+π/2)=1/5
所以√2sin(x+π/2)=√2cosx=1/5,cosx=√2/10
于是sinx=√98/10

用两角和公式将上述两式展开
sin(x+π/4)=4/5即 二分之根号二cosx+二分之根号二sinx=4/5
cos(x+π/4)=-3/5即 二分之根号二cosx-二分之根号二sinx=-3/5
两式相减得 √2sinx=8/5
sinx=2√2/5

√2/2(sinx+cosx)=4/5
√2/2(cosx-sinx)=-3/5
√2sinx=7/5
sinx=7√2/10