Question

What is the formula for the distance between two polar coordinates?

Answer 1

The distance is $\sqrt{r_1^2+r_2^2-2r_1{r}_{2}cos({\theta }_1-{\theta }_2)}$ if we are given $P_1=(r_1,{\theta }_1)$ and $P_2=(r_2,{\theta }_2)$
This is an application of the cosine law. Taking the difference between ${\theta }_1$ and${\theta }_2$ gives us the angle between side $r_1$ and side $r_2$. And the cosine law gives us the length of the 3rd side.