Question

Find sin theta if cot theta = -2 and cos theta < 0

Answer 1

cot(θ) = cos(θ)/sin(θ)

So if both cot(θ) and cos(θ) are negative, that means sin(θ) must be positive.

Recall that

cot²(θ) + 1 = csc²(θ) = 1/sin²(θ)

so that

sin²(θ) = 1/(cot²(θ) + 1)

sin(θ) = 1 / √(cot²(θ) + 1)

Plug in cot(θ) = -2 and solve for sin(θ) :

sin(θ) = 1 / √((-2)² + 1)

sin(θ) = 1/√(5)