1.已知sina=2/3,cosβ=-3/4,且a,β都是第二象限角,求sin(a-β),cos(a+β),tan2a.

Question

1.已知sina=2/3,cosβ=-3/4,且a,β都是第二象限角,
求sin(a-β),cos(a+β),tan2a.

Answer 1

cosa=√(1-sin^2a)=√(1-4/9)=-√5/3
sinβ=√(1-cos^2β)=√(1+9/16)=√7/4
sin(a-β)=sinacosβ-cosasinβ
=(2/3)(-3/4)-(-√5/3)(√7/4)
=(-1/2)+(√35/12)
=(-6+√35)/12
cos(a+β)=cosacosβ-sinasinβ
=(-√5/3)(-3/4)-(2/3)(√7/4)
=(3√5/12)-(2√7/12)
=(3√5-2√7)/12
tana=-2√5/5
tan2a=2tana/(1-tan^2a)
=[-4√5/5]/[1-4/5]=-4√5