Given:
A circle's arc segment of length = 1, and the Y-coordinate of its endpoint
The Y-coordinate of the circle's center
The fact that the circle goes through the origin: (0,0)
Is it possible to find the X-coordinate of the arc segment's endpoint?
I refer to the image below:
The arc segment is the thick blue segment on the black dotted circle: its length is 1, the horizontal coordinate of its endpoint is marked by the upper dotted blue line, and the blue dot on the Y-axis. The circle center's vertical coordinate is the bottom blue dotted line, and the required coordinate is the red dot on the X-axis, marked by a red dotted line: this is what I am looking for.
Much obliged!
According the diagram, the following can be established,
$$\theta r = 1,\>\>\>\>\>b=r\sin(\pi-\theta),\>\>\>\>\>-x=r+r\cos(\pi-\theta)$$
which lead to,
$$x=-b\tan\frac\theta2,\>\>\>\>\>b=\frac{\sin\theta}{\theta}$$
Eliminate θ to obtained the equation
$$\tan^{-1}\frac xb = \frac x{b^2+x^2}$$
Therefore, the $x$-coordinate of the endpoint can be obtained if $b$, the $y$-coordinate of the endpoint is known.