Find cot theta/2 if tan theta = -7/24 and 3pi/4 < theta < 2pi.

Answer 1

When we talk of tan(θ) being -7/24 and in quadrant 4, we are talking about this red triangle:

We can then calculate the hypotenuse using the Pythagorean Theorem to be sqrt((-7)² + 24²) = 25

Now we can define cos(θ) as 24/25 and sin(θ) as -7/25

Now, we can use the half angle equation to find the tan(θ/2).

tan(θ/2) = (1 - cos(θ))/sin(θ) = (1 - 24/25)/(-7/25) = (1/25)/(-7/25) = -1/7

Since cot = 1/tan then cot(θ) = 1/(-1/7) = -7