Question

Find sin θ if cot θ = - 4 and cos θ < 0.

Answer 1

sin θ = 0.2426.

Explanation:

The cot and cos are both negative in the second quadrant.

cot θ = 1 / tan θ = -4 so tan θ = -1/4.

θ = 165.96 degrees

sin θ = 0.2426.

Answer 2

sin θ = √17 / 17

Explanation:

Pythagorean identity:

1 + cot² θ = csc² θ

1 + (-4)² = csc² θ

17 = csc² θ

csc θ = ±√17

sin θ = ±1 / √17

cot θ < 0 and cos θ < 0, so θ is in the second quadrant. Therefore, sin θ > 0.

sin θ = 1 / √17

Or, as a proper fraction:

sin θ = √17 / 17