Please see the attached thumbnails. I have here this lovely archimedean spiral inscribed within a unit circle and I want to derive angle in terms of angle . is the angle under the pink line, is the angle under the spiral radius, green line.

Known: and the equation for the spiral radius which is just the polar equation for said spiral curve, and b, the y intercept of this curve:

Again, the spiral is inscribed within a unit circle. I want to express in terms of .

(If can be solved for in terms of that would be fine too.)

So far I’ve taken the following approach:

1) I convert the curve from polar to Cartesian form:

2) I find the intercept between the spiral and the line by substituting in for y:

Which simplifies to:

3) This, then, is equal to , so we can express in terms of :

But, alas, the reverse is not so simple. I cannot find a way to solve this equation for in terms of .

Perhaps a whole different approach is necessary. Your help is much appreciated.