In taking a double integral in polar coordinates, I'm learning that we can break up the surface into little polar rectangles with $\Delta r$ and $\Delta \theta$.
Therefore, the Riemann sum of a function $f(x,y)$ associated with such a polar partition is $$\sum_{i=1}^k f(r_i^*\cos \theta_i^*, r_i^*\sin \theta_i^*)\Delta A_i$$ where the asterisks connote the average value for those variables in the given infinitesimal polar rectangle.
That makes sense to me. What I'm not getting is the following step, in which we break it up further into
$$\sum_{i=1}^k f(r_i^*\cos \theta_i^*, r_i^*\sin \theta_i^*)r_i^* \Delta r \Delta \theta$$
such that $dA = rdrd\theta.$ It seems to me that the distance over $\Delta \theta$ would be $$r\sin{\Delta\theta},$$ because it's approximately the "opposite" side of a near-right triangle with $\theta$ determining its length accordingly.
Why is it just $r\Delta\theta$, not involving the $\sin$? Is there some obvious geometry that makes it clear once visualized?