数学题在线解答已知0<α<π/2<β<π,cos(β-π/4π)=1/3,sin(α+β)=4/5.求cos(α+π/4)的值
问题描述:
数学题在线解答已知0<α<π/2<β<π,cos(β-π/4π)=1/3,sin(α+β)=4/5.求cos(α+π/4)的值
答
学习一下。
答
由题意-3π/4<π/4-β<-π/4,π/2<α+β<3π/2
又cos(π/4-β)=1/3,∴sin(π/4-β)=-2√2/3
sin(α+β)=4/5,∴cos(α+β)=-3/5
∴cos(α+π/4)=cos((π/4-β)+(α+β)) 两角和余弦展开
=(8√2-3)/15
答
设β-π/4为A,α+β为B,则α+π/4为B-ACOS(α+π/4)=COS(B-A)=COSBCOSA+SINBSINACOSA=1/3,SIN²A=8/9.且π/4<A<3/4π,所以SINA=2√2/3SINB=4/5,COS²B=9/25,且π<B<3π/2,所以COSB=-3/5原式=-3/5×1...