What is the length of the cardioid $r=1-\cos(\theta)$?

Question

I know generally how to solve this problem and was able to solve it a week or so ago. However, I keep getting stuck when trying to find $dr/d\theta$. I know that it should simplify to $-\sin(\theta)$ from my notes but I can't get anything other than (1-cos(theta))/sin(theta) when I try now.

Answer 1

We want $$\int_0^{2\pi} \sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}\,d\theta.$$ In our case, we want to integrate $\sqrt{(1-\cos\theta)^2+\sin^2\theta}$, which simplifies to $\sqrt{2-2\cos\theta}$.

Now use the trigonometric identity $\cos \theta=1-2\sin^2(\theta/2)$.