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
