Question

已知cos(x+兀/4)=3/5，兀/2<x<3兀/2求cos(2x+兀/4)的值

Answer 1

cos(x+π/4)=3/5
cosxcosπ/4-sinxsinπ/4=3/5
cosx-sinx=3根号2/5
(cosx-sinx)^2=(3根号2/5)^2
cos^2x+sin^2x-2sinxcosx=18/25
1-sin2x=18/25
sin2x=7/25

∵π/2<x<3π/2
又：cos(x+π/4)=3/5＞0
∴3π/2＜x+π/4＜2π，即，5π/4＜x＜7π/4
又：π/2<x<3π/2
∴5π/4<x<3π/2
∴5π/2＜2x＜3π
∴2x为第二象限角
∴cos2x=-根号(1-sin^2(2x))=-24/25
∴cos(2x+π/4)=cos2xcosπ/4-sin2xsinπ/4
=根号2/2(cos2x-sin2x)
=根号2/2(-24/25-7/25)
=-31根号2/50