Question

已知cos(a+兀/4)=3/5,兀/2小于或等于a小于3兀/2,求cos（2a+兀/4)的值.

Answer 1

cos(a+兀/4)=3/5
根号2/2*cosa-根号2/2*sina=3/5
cosa-sina=3根号2/5
cosa=3根号2/5+sina
cosa^2+sina^2=1
(3根号2/5+sina)^2+sina^2=1
18/25+2sina^2+6根号2sina/5=1
2sina^2=7/25-6根号2sina/5
2sina^2+6根号2sina/5-7/25=0
sina=[-6根号2/5+根号(72/25+56/25)]/4=[-6根号2/5+8根号2/5]/4=根号2/10
或sina=[-6根号2/5-根号(72/25+56/25)]/4=[-6根号2/5-8根号2/5]/4=-7根号2/10
因为兀/2小于或等于a小于3兀/2
所以sina=-7根号2/10
cos（2a+兀/4)=根号2/2*cos2a-根号2/2*sin2a=根号2/2-根号2sina^2-根号2sinacosa
=根号2/2-根号2sina^2-根号2sina(3根号2/5+sina)
=根号2/2-根号2sina^2-6sina/5-根号2sina^2
=-2根号2sina^2-6sina/5+根号2/2
=-根号2(7/25-6根号2sina/5)-6sina/5+根号2/2
=12sina/5-7根号2/25-6sina/5+根号2/2
=6sina/5+11根号2/50
=6(-7根号2/10)/5+11根号2/50
=-42根号2/50+11根号2/50
=-31根号2/50