I don't understand how you find the function value using coordinate points and give exact answers, like cot 7pi/6. I know how to them in decimals because you just put them in the calculator.
From the 30-60 right triangle you know the exact values of sin and cos of 30 deg, or pi/6 (the values are 1/2 and sqrt(3)/2), and sin and cos of 60 deg, or pi/3 (the values are sqrt(3)/2 and 1/2).
From a 45-45 right triangle you know that the exact values of sin and cos of 45 deg, or pi/4, are both sqrt(2)/2.
Using these values (and 0 and 1 for the angles 0, pi/2, pi, 3 pi/2 and 2 pi), and negatives where needed, you label the sin and cos of every angle around the unit circle which is a multiple of pi/6, and every angle which is a multiple of pi/4. These values are all exact.
Having done that you use the definitions of the tan, cot, sec and csc functions to find the values of those functions at the angles you have labeled.
For example if you want cot(7 pi/6), you first see from the labeled picture that sin(7 pi / 6) = -1/2 and cos(7 pi/6) = -sqrt(3)/2.
You know that cot(theta) = cos(theta) / sin(theta). So cot(7 pi/6) = cos(7 pi / 6) / sin(7 pi / 6) = ( -sqrt(3) / 2 ) / (-1/2) = (sqrt(3)/2) * (2/1) = sqrt(3).
If you wanted to find the tan, csc or sec you would use those definitions, all of which involve the sine and/or cosine functions.