已知6sin^2α+sinαcosα-cos^2α=0,求sin(2α+π/3)的值

题目

已知6sin^2α+sinαcosα-cos^2α=0,求sin(2α+π/3)的值

答案

sin(2α+π/3)=3*(根号3)-4/10或4*(根号3)+3/10
6sin^2α+sinαcosα-cos^2α=0
(2sinα+cosα)(3sinα-cosα)=0
2sinα+cosα=0或3sinα-cosα=0
tanα=-1/2或tanα=1/3
sin(2α+π/3)=1/2*sin2α+(根号3)/2*cos2α
用公式sin2α=2tanα/1+tan^2α cos2α=1-tan^2α/1+tan^2α可求得sin2α cos2α
tanα=-1/2时 sin2α=-4/5 cos2α=3/5 sin(2α+π/3)=3*(根号3)-4/10
tanα=1/3时 sin2α=3/5 cos2α=4/5 sin(2α+π/3)=4*(根号3)+3/10