If you plot these two relations:
$$y=cos(x)$$ $$x^2 + y^2 = 1$$
The cosine wave seems to "hug" the unit circle shown below.
I'm wondering the calculus explanation for this, as I believe it must have to do with the relationship between the implicit derivative of the unit circle, where the slope of the tangent line at any point is found using the negative cotangent function. But this doesn't seem to explain why the unit circle is wrapped around the cosine function like this as the derivative is asymptotic at $\frac{\pi}{2}$. And even then, is there a geometric intuition for why they behave so similarly around 0?