Start from the definitions:
#csc(x)=1/sin(x)#; #sec(x)=1/cos(x)#;
#tan(x)=sin(x)/cos(x)#; #cot(x)=cos(x)/sin(x)#
Based on this, all we need to define using the unit circle are #sin(x)# and #cos(x)#.
By definition, #sin(x)# is an ordinate (Y-coordinate) and #cos(x)# is an abscissa (X-coordinate) of a point lying on a unit circle at the end of a radius that forms an angle #x# radians with the positive direction of the X-axis (counterclockwise from X-axis to this radius).
Using all the above, let's, for example, find #sec(5pi/6)#.
#sec(5pi/6)=1/cos(5pi/6)#
Angle #5pi/6=150^0# in a unit circle is determined by a radius from an origin of coordinates #O# to a point #A# in the second quadrant such that an angle #∠XOA=5pi/6#. Drop a perpendicular from point #A# on the X-axis. Its base, point #B#, has a coordinate #-sqrt(3)/2#. This is obvious from the triangle #ΔOAB#. We can conclude that abscissa of point #A# equals to #-sqrt(3)/2#.
Therefore,
#cos(5pi/6)=-sqrt(3)/2#
From this we find
#sec(5pi/6)=-2/sqrt(3)=-2sqrt(3)/3#
