Use the information given about the angle θ, 0 ≤ θ ≤ 2π, csc θ = -3/2 , tan θ > 0 Find cos θ/2

Question

Use the information given about the angle θ, 0 ≤ θ ≤ 2π, to find the exact value of the indicated trigonometric function.

csc θ = - , tan θ > 0 Find cos .

Answer 1

sinθ=-⅔ and tanθ>0 means that θ is in the third quadrant π<θ<3π/2 and π/2<θ/2<3π/4.

cosθ=-√1-4/9=-√(5/9)=-√5/3, because cosine<0 in Q3.

cosθ=2cos²(θ/2)-1, cos(θ/2)=-√((1+cosθ)/2)=-√((1-√5/3)/2)=-0.3568 approx, because θ/2 is in Q2 where cosine<0.