I'd like to calculate the following angle, $\phi$ between the center of a circle $O$ and a point on the circumference $Q$ (wrt the $x$ axis), where $Q$ is determined by 2 parameters, $a$ and $\theta$, as defined in the diagram below. ($a$ is given as the fraction of the radius.)
Would anyone be able to help determine an analytic expression for $\phi$ in terms of $a$ and $\theta$?
Would be extremely grateful for any help. Thank you.
As in @StephenDonovan comment, consider $\triangle OPQ $ and apply the law of sines
$ \dfrac{ \sin(\theta) } {r} = \dfrac{ \sin(\theta - \phi) }{r - a } $
From which it follows immediately that
$ \phi = \theta - \sin^{-1}\left( \dfrac{(r - a) \sin(\theta) }{r } \right) $