How would you find a negative radian on the unit circle? this is so confusing to me.

example: the cot of -pi/6

Here, As we are dealing with negative radian, we will go clockwise instead of counter-clock wise which we usually do in positive cases, 

So, Terminal values are, 0, -π/2 , -π, -3π/2, 

Now, You have: Cot (-π/6)

Multiply terminal values by 3/3, it would be: 0, -3π/6, -π, -9π/6
So, we can figure out, -π/6 is in between 0 and -3π/6, 

More nearer to 0, so reference value = 0 - (-π/6) = Cot (π/6)
We know, Cot (π/6) = √3
As quadrant is 4th which is negative for Cot, it would be -√3

In short, Your Final Answer would be -√3

Hope this helps!