When the circle is centered at the origin and the angle of interest (α) is measured counterclockwise from the positive x-axis, the trig functions can be determined from the coordinates of the point (x, y) and the radius of the circle (r) (which can also be determined from the coordinates of the point).
The radius is
r = √(x² +y²)
The trig functions are (remember SOH CAH TOA?)
sin(α) = y/r
cos(α) = x/r
tan(α) = y/x
These ratios hold regardless of quadrant.