Question

已知：0< a <派/2< β<派，cos ( β+派/4）=1/3,sin (a+ β)=4/5

已知：0< a <派/2< β<派，cos ( β+派/4）=1/3,sin (a+ β)=4/5
1。求sin2 β的值
2。求cos (a +派/4）的值

Answer 1

cos（π/2+α）= -sinα

cos ( β+π/4）=1/3,
sin (a+ β)=4/5
cos ( β+π/4）=1/3
cos (2β+π/2)=2cos²(β+π/4)-1
cos (2β+π/2)=2*(1/3)²-1
cos (2β+π/2)=2/9-1
cos (2β+π/2)=-7/9
-sinα2β=-7/9
sinα2β=7/9

0< a <π/2< β<π
π/2< β<π
3π/4<β+π/4<5π/4
cos ( β+π/4)=1/3
cos ( β+π/4)<0
题目有问题

Answer 2

解：∵cos(β+π/4)=cosβcos(π/4)-sinβsin(π/4)
=(√2/2)(cosβ-sinβ)
又cos(β+π/4）=1/3
∴(√2/2)(cosβ-sinβ)=1/3
变形得：cosβ-sinβ=√2/3

两边同时平方得：(1/2)(cos^2β-2sinβcosβ+sin^2β）=1/9
整理得：sin2β=7/9
∵0<α<π/2<β<π
∴π<2β<2π，π/2<α+β<3π/2
∴sin2β=-7/9
又sin(α+β)=4/5
∴cos(α+β）=-√1-sin^2(α+β)
=-√1-(4/5)^2
=-3/5
sin(α+β)=sinαcosβ+sinβcosα=4/5
cos(α+β)=cosαcosβ-sinαsinβ=-3/5
两式相减，得：sinαcosβ+sinβcosα-cosαcosβ+sinαsinβ=7/5

因有事，余下的你自己做一下。