Question

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

Answer 1

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.

Answer 2

The cos of 5pi/3 is in radians, so we have to convert it to degrees. To do this, we’ll multiply 5pi/3 by 180/pi.
This gives us the new equation of 900pi/3pi. The pis cancel out, leaving 900/3=300 degrees.

This angle is 300 degrees, so we know that it’s 60 degrees away from the 0 or 360 degree mark on the unit circle. This means that we have to take the cos of 60 degrees, which is .5.

The unit circle, in the shown picture, shows that 300 degrees would be in the in the fourth, or bottom right, quadrant. Since cos values in the fourth quadrant are always positive, we know that the cos of 300, or in other words cos of 60, is positive.

Therefore, the final answer would be 0.5 in Quadrant 4.