Explain how to find the exact value of cot 5π/3, including quadrant location.

Start from the positive x-axis and move counterclockwise about the origin.  You will first pass through Quadrant I, then Q II, then Q III, and so on.  Convince yourself that the angle pi/3 is in Q I; 2 pi/3 is in Q II; 5 pi/3 is in Q IV.  Because 5 pi/3 is in Q IV, its adjacent side is positive and its opposite side is negative.

                     adj         +1                    1             sqrt(3)            1            sqrt(3)
cot 5pi/3 = --------- = ------------- = - ----------- = ------------- * ----------- = ------------
                     opp        -sqrt(3)          sqrt(3)        sqrt(3)       sqrt(3)             3

Note:  If this is not completely clear to you, look up "special triangles" and look for a picture showing the hypotenuse, the adjacent side and the opposite side of the central angle.  This is a "30-60-90 degree triangle" which has sides 1 and sqrt(3) and hypotenuse 2.