Tangent to the circle given a point it must pass through a point on another circle

Question

I need to find the angle at origin caused by two lines (one is the radius of one circle, and the other is tangent to the other circle).

Please see image below:

The Point A on the green circle is known, the dashed line and shorter full line begin at the origin. The other line is tangent to the red circle and passing through Point A.

How would I calculate the angle a?

I know this is probably a basic question but my head is burnt out at work and I just need reassurance before making an epic mistake.

Answer 1

Since from the drawing it is clear that $\;r<\sqrt 2\;$, we have that if $\;\alpha\;$ is the wanted angle then

$$\cos\alpha=\frac r{\sqrt2}\implies \alpha=\arccos\frac r{\sqrt2}$$