CPM Homework Help : PC Problem 8-78

Find the exact value of $\cot θ$ given $\cos θ = \frac { \sqrt { 3 } } { 2 }$ and $\csc θ \lt 0$ .

Hint 1:

Since the $\csc\left(θ\right) \lt 0$ , it must be in Quadrants III or IV.
Since the $\cos\left(θ\right) \gt 0$ , it must be in Quadrant IV.

Hint 2:

Sketch a diagram and label.

Diagram:

Hint:

$\text{cot}(\theta)=\frac{\text{adjacent side}}{\text{opposite side}}$

Circle with point in fourth quadrant, segment from center to point, is hypotenuse of right triangle, whose vertical leg labeled negative 1, horizontal leg labeled square root of 3, hypotenuse labeled 2.