Find the exact value of cot(θ) for an angle θ with sin(θ) = -1/6 and with its terminal side in Quadrant III.
Being in Quadrant III, the x-values and y-values are both negative.
If you draw the reference triangle, the y-segment is -1 when the hypotenuse is 6. (Because sin(theta) = -1/6.)
Using the Pythagorean Theorem, we can find that the x-segment is - sqrt(35).
So the tan(theta) = -1 / -sqrt(35),
which makes the cot(theta) = sqrt(35).