Question

已知向量a=(cosα,sinα),b=(sinβ,cosβ),β属于(0,π),tanβ/2=1/2,向量ab=5/1

Answer 1

tanβ= 2*tan(β/2)/[1-tan²(β/2)]

=2*1/2/[1-(1/2)²]

=1/(3/4)

=4/3

β属于(0,π)

sinβ=4/5

cosβ=3/5

ab=5/13

cosαsinβ+sinαcosβ=ab

cosαsinβ+sinαcosβ=5/13

sin(α+β)=5/13

cos(α+β)=±12/13

当cos(α+β)=12/13时

sinα

=sin[(α+β)-β]

=sin(α+β)cosβ-cos(α+β)sinβ

=5/13*3/5-12/13*4/5

=15/64-48/65

=-33/65

当cos(α+β)=-12/13时

sinα

=sin[(α+β)-β]

=sin(α+β)cosβ-cos(α+β)sinβ

=5/13*3/5-(-12/13)*4/5

=15/64+48/65

=63/65