In the Cartesian plane let $O(0,0)$ , $P(\cos A,\sin A)$, $Q( - \sin A , \cos A)$ be the vertices of a triangle. Two circles are drawn with $OP$, $OQ$ as diameter intersecting at $R$. Then find $OR$.
I know that $Q$ is the reflection of $P$ in $y$-axis. However I have not been able to come up with anything substantial. Please help me solve this.
It's easy with pure geometry: the right triangle $OPQ$ is isosceles, and $OR$ is an altitude of this right triangle, hence a perpendicular bisector of the hypotenuse $PQ$. As a result, $OR^2=\dfrac12$, and the polar angle of $R$ is $A+\dfrac\pi4$.