How do I find the "unwinding" rate of a polar equation of two variables?

Question

I was playing around with an online calculator and polar functions, seeing what kind of function transformations I could do with a slider. I had the equation: $$ r =\cos(\sin(\tan(\frac{\theta}{c}))) $$ where c was constant. Then, I clicked the "play" button on c, making c increase by 0.05 every frame, and found some sort of unwinding behavior, as seen here. I also found that the farther $c$ was from 0, the slower it unwound. I feel like doing a derivative but I don't know how that applies to my problem.
My Question:
How do I find the "unwinding rate" of a polar function of two variables?
Thank you in advance,
Jonah