Question

若sinα-sinβ=1-√3/2,cosα-cosβ=1/2,则cos(α-β)的值为多少?

若sinα-sinβ=1-√3/2,cosα-cosβ=1/2,则cos(α-β)的值为多少?

Answer 1

sinα-sinβ=1-√3/2,
cosα-cosβ=1/2
两边平方：
sin^2α-2sinαsinβ+sin^2β=7/4-√3
cos^2α-2cosαcosβ+cos^2β=1/4
相加
2-2（sinαsinβ+cosαcosβ）=2-√3
所以cos(α-β)=√3/2

Answer 2

sinα-sinβ=1-√3/2
(sinα-sinβ)^2=(1-√3/2)^2
sina^2-2sinasinb+sinb^2=(4-2√3)/4=1-√3/2

cosα-cosβ=1/2
(cosα-cosβ)^2=1/4
cosa^2-2cosacosb+cosb^2=1/4

[sina^2-2sinasinb+sinb^2]+[cosa^2-2cosacosb+cosb^2]=1-√3/2+1/4
2-2(sinasinb+cosacosb)=5/4-√3/2
2(sinasinb+cosacosb)=√3/2+1/4
cos(a-b)=√3/4+1/8