Solution for finding polar coordinates given arc length

Question

What is the general theorem and/or proof that allows one to deduce the X and Y values of a point that lies on a circle if given the arc length from another point on the same circle?

I made a quick picture in Paint to demonstrate my question better.

Given coordinates of the blue point, and an arc length (in red), how can one determine the coordinates of the yellow point?

Answer 1

Given $(x_b,y_b)$, the coordinates of the blue point, the $y_b$ value is the radius of the circle.

$y_b=r$

So now find the angle (in radians)

$arc length=r\theta_b$

$\theta_b=(arc length)/y_b$

$\pi/2-\theta_b=\theta_y$

Then use basic trig equations (cos and sin) with the radius $r=y_b$ to find the coordinates of the yellow point.