I know the radius "r" of a circle. I have a point "P", always in the circle and always "looking" at the center of the circle, with a certain angle, or overture "a". I know the distance between "P" and the center of the circle. I would like to know the angle or overture "b" from the center, so that "b" covers the same arc of the circle as "a".
Here's a schema explaining the problem :
The goal is to retrieve the angle "b" from all the other parameters.
Thanks a lot in advance !
Best Answer
Let $\alpha=a/2$ and $\beta=b/2$. Applying the sine law to the triangle with sides $d$, $r$ we get: $$ {r\over\sin\alpha}={d\over\sin(\beta-\alpha)}, $$ which after expanding $\sin(\beta-\alpha)$ becomes: $$ \sin\beta=\tan\alpha\cos\beta+{d\over r}\tan\alpha. $$ This equation can be solved, for example, plugging it into $\sin^2\beta+\cos^2\beta=1$ and solving for $\cos\beta$: $$ \cos\beta=\cos\alpha\sqrt{1-{d^2\over r^2}\sin^2\alpha}-{d\over r}\sin^2\alpha, $$ where I discarded the negative solution as $0\le\beta\le\pi/2$.
EDIT.
Here's a graph of $b$ vs. $d/r$, comparing (for $a=180°$) the exact solution above (black curve) with the approximate solution $b=(1+d/r)a$ (red curve). The difference is less pronounced for smaller values of $a$.