# What is the distance between the following polar coordinates?:  (3,(-7pi)/12), (2,(7pi)/8)

$\sqrt{13 - 12 \cos \left(\frac{35 \pi}{24}\right)} \approx 3.81658411497$
Say you have two polar coordinates $\left({r}_{1} , {\theta}_{1}\right)$ and $\left({r}_{2} , {\theta}_{2}\right)$. In this case make a triangle with vertices at the origin and each point. Around the vertex of the origin we have two sides of lengths ${r}_{1}$ and ${r}_{2}$, and we have angle $| {\theta}_{1} - {\theta}_{2} |$. By the law of cosines, the third side is $\sqrt{{r}_{1}^{2} + {r}_{2}^{2} - 2 {r}_{1} {r}_{2} \cos \left({\theta}_{1} - {\theta}_{2}\right)}$
$\sqrt{13 - 12 \cos \left(\frac{35 \pi}{24}\right)} \approx 3.81658411497$