题目

已知cos(π/4+a)=3/5,且7/12π提问时间：2020-11-03

答案

解.∵7π/12∴5π/6<π/4+a<2π
∵cos(π/4+a)=3/5>0
∴3π/2<π/4+a<2π
∴sin(π/4+a)=-4/5,tan(π/4+a)=-4/3
∴sina=sin(a+π/4-π/4)=sin(a+π/4)cosπ/4-cos(a+π/4)sinπ/4
=(-4/5)*(√2/2)-(3/5)*(√2/2)=-7√2/10
cosa=cos(a+π/4-π/4)=cos(a+π/4)cosπ/4+sin(a+π/4)sinπ/4
=(3/5)*(√2/2)+(-4/5)*(√2/2)=-√2/10
∴tana=sina/cosa=7
sin(2a)+2sin²a=2sinacosa+2sin²a
=2(-7√2/10)*(-√2/10)+2(7√2/10)²
=7/25+49/25=56/25
∵tan(a+π/4)=(1+tana)/(1-tana)
∴1-tana=(1+tana)/tan(a+π/4)=(1+7)/(-4/3)=-6
∴(sin(2a)+2sin²a)/(1-tana)=(56/25)/(-6)=-28/75