Question

高中数学必修四（和角公式）已知sinα+sinβ=3/5,cosα+cosβ=4/5,求cos（α-β）的值

Answer 1

cos(α-β)=cosαcosβ+sinαsinβ

(sinα+sinβ)^2=(sinα)^2+(sinβ)^2+2sinαsinβ

所以2sinαsinβ=9/25-(sinα)^2-(sinβ)^2

同理2cosαcosβ=16/25-(cosα)^2-(cosβ)^2

cos(α-β)=1/2(2cosαcosβ+2sinαsinβ)

=1/2[9/25-(sinα)^2-(sinβ)^2+16/25-(cosα)^2-(cosβ)^2]

=1/2{(9/25+16/25)-[(sinα)^2+(cosα)^2]-[(sinβ)^2+(cosβ)^2]}

因为(sinα)^2+(cosα)^2=1,(sinβ)^2+(cosβ)^2=1

所以=1/2[1-1-1]=-1/2

(^2代表平方）