已知cos(a+π/4)=3/5, π/2≤a≤3π/2, 求cos(2a+π/4)的值

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已知cos(a+π/4)=3/5, π/2≤a≤3π/2, 求cos(2a+π/4)的值

已知cos(a+π/4)=3/5, π/2≤a≤3π/2, 求cos(2a+π/4)的值
已知cos(a+π/4)=3/5, π/2≤a≤3π/2, 求cos(2a+π/4)的值

已知cos(a+π/4)=3/5, π/2≤a≤3π/2, 求cos(2a+π/4)的值
π/2≤a≤3π/2
cos(a+π/4)=3/5>0
a+π/4>3π/2
3π/2

由1/2<3/5<√2/2且π/2≤α≤3π/2得
5π/3<α+π/4<7π/4,即17π/12<α<3π/2
cos(α+π/4)=cosαcos(π/4)-sinαsin(π/4)
=(√2/2)(cosα-sinα)=3/5
故cosα-sinα=3√2/5,平方得
cos²α-2cosαsinα+sin²α=18/25...

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由1/2<3/5<√2/2且π/2≤α≤3π/2得
5π/3<α+π/4<7π/4,即17π/12<α<3π/2
cos(α+π/4)=cosαcos(π/4)-sinαsin(π/4)
=(√2/2)(cosα-sinα)=3/5
故cosα-sinα=3√2/5,平方得
cos²α-2cosαsinα+sin²α=18/25,即
sin2α=7/25,再由17π/6<2α<3π,得cos2α=-24/25,故
cos(2α+π/4)=cos2αcos(π/4)-sin2αsin(π/4)
=(√2/2)(cos2α-sin2α)=-31√2/50

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31√2/50