There is a two variable function, called $\text{atan2}$ in C, that may do the job for you, if something like it is built into the piece of software that you are using.
For some discussion of the $\text{atan2}$ function, see this.
Roughly speaking, $\text{atan2}(y,x)$ is $\arctan(y/x)$ if $x$ is positive. If $x$ is negative, and $y\ge 0$, then $\text{atan2}(y,x)=\pi+\arctan(y/x)$, while if $x<0$ and $y<0$, then $\text{atan2}(y,x)=-\pi+\arctan(y/x)$. And so the program won't blow up, $\text{atan2}(y,x)$ is defined in the reasonable way when $x=0$.
In particular, $\text{atan2}(1/\sqrt{2},-1/\sqrt{2})=3\pi/4$, precisely what you wanted. You may be less happy with $\text{atan2}(-1/\sqrt{2},-1/\sqrt{2})$.
Warning: While many software packages implement an $\text{atan2}$-like function, the name and the syntax are not universal. Sometimes $x$ and $y$ are interchanged. The details for Fortran, C, Mathematica, MATLAB, and Excel, to mention some examples, are slightly different!
If your first point on the circle (lying on the line parallel to the $x$-axis) is $P_0=(O_x+r,O_y)$ then the next point (moving counter-clockwise) is $P_1=(O_x+r\cos(\frac{360}{a}),O_y-\sin(\frac{360}{a}))$.
Each time the angle increases by $\frac{360}{a}$ degrees, so more generally:
$$P_n=(O_x+r\cos\left (\frac{360n}{a}\right), O_y-r\sin\left (\frac{360n}{a}\right))$$ for $0\leq n\leq a-1$
$r$, with angle $\theta$ to the $y$-axis">
Best Answer
From the picture, it seems that your circle has centre the origin, and radius $r$. The rotation appears to be clockwise. And the question appears to be about where the point $(0,r)$ at the top of the circle ends up.
The point $(0,r)$ ends up at $x=r\sin\theta$, $y=r\cos\theta$.
In general, suppose that you are rotating about the origin clockwise through an angle $\theta$. Then the point $(s,t)$ ends up at $(u,v)$ where $$u=s\cos\theta+t\sin\theta\qquad\text{and} \qquad v=-s\sin\theta+t\cos\theta.$$