Question

已知α∈（π/2，π），sinα=根号下5/5.(1)求sin（α+π/4）的值；（2）求cos（5π/6-2a）的值.

Answer 1

解：∵α∈(π/2,π)
∴cosα<0
∵sinα=√5/5
∴cosα=-√[1-(sinα)^2]=-2√5/5
故（1）sin(α+π/4)=sinαcos(π/4)+cosαsin(π/4)
=(√5/5)(√2/2)+(-2√5/5)(√2/2)
=-√10/10
（2）cos(5π/6-2α)=cos[π-(π/6+2α)]
=-cos(π/6+2α)
=-cos(π/6)cos(2α)+sin(π/6)sin(2α)
=-(√3/2)[(cosα)^2-(sinα)^2]+(1/2)(2sinαcosα)
=-(√3/2)[(-2√5/5)^2-(√5/5)^2]+(1/2)[2(√5/5)(-2√5/5)]
=3√3/10-2/5。