The endpoint of an angle #x# lies in the 4th quadrant. Here is why.
First of all, let's recall a few definitions.
Recall the definition of a trigonometric function #sec(x)#:
#sec(x) =# (by definition) #=1/cos(x)#
Recall the definition of a trigonometric function #cot(x)#:
#cot(x) =# (by definition) #=cos(x)/sin(x)#
Recall the definition of a trigonometric function #cos(x)#:
Consider a unit circle around the origin of coordinate #O# and a point #A# on this circle such that an angle from the positive direction of the X-axis #OX# counterclockwise to a ray #OA# equals to #x# (usually, radians). Then the abscissa (X-coordinate) of the point #A# is, by definition, #cos(x)#.
Finally, recall the definition of a trigonometric function #sin(x)#:
The same arrangement as above for #cos(x)#, but in this case the ordinate (Y-coordinate) of the point #A# is, by definition, #sin(x)#.
Since #sec(x)=1/cos(x)>0#, #cos(x)>0#, which means that abscissa of a position of the endpoint of an angle #x# is positive.
Since #cot(x)=cos(x)/sin(x) < 0# and #cos(x)>0#, #sin(x)# must be negative, which means that ordinate of a position of the endpoint of an angle #x# is negative.
So, the endpoint of an angle #x# has positive abscissa (X-coordinate) and negative ordinate (Y-coordinate), which puts it into 4th quadrant.
