Use identities to find values of sin 2θ and cos 2θ for sin θ = 2/5, cos θ < 0

Question

Could anyone help me solve this problem? I've been stuck on it for a while.

Answer 1

sinθ=2/5.  cos²θ=1-sin²θ gives cos²θ=1-4/25=21/25

cosθ<0 gives cosθ=-(√21)/5

sin2θ=2sinθcosθ=2×(2/5)×(-(√21)/5)=-4(√21)/25

cos2θ=cos²θ-sin²θ=21/25-4/25=17/25

 

Answer 2

Given the sine value, we can infer that the associated cosine value is -√21/5, using the Pythagorean identity.

 

Now use the double angle formulas: sin 2θ = 2 sinθcosθ and cos 2θ = cos²θ-sin²θ