# What is the distance between the following polar coordinates?: <mstyle displaystyle="true">

What is the distance between the following polar coordinates?: $\left(7,\frac{5\pi }{4}\right),\left(3,\frac{13\pi }{8}\right)$
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Step 1
using the polar version of the distance formula
${d}^{2}={r}_{1}^{2}+{r}_{2}^{2}-\left[2{r}_{1}{r}_{2}\mathrm{cos}\left({\theta }_{2}-{\theta }_{1}\right)\right]$
Let $\left({r}_{1},{\theta }_{1}\right)=\left(7,\frac{5\pi }{4}\right)\phantom{\rule{1ex}{0ex}}\text{and}\phantom{\rule{1ex}{0ex}}\left({r}_{2},{\theta }_{2}\right)=\left(3,\frac{13\pi }{8}\right)$
Step 2
$⇒{d}^{2}={7}^{2}+{3}^{2}-\left[2×7×3\mathrm{cos}\left(\frac{13\pi }{8}-\frac{5\pi }{4}\right)\right]$
${d}^{2}=49+9-\left(42\mathrm{cos}\left(\frac{3\pi }{8}\right)\right)$
$⇒d=\sqrt{58-42\mathrm{cos}\left(\frac{3\pi }{8}\right)}\approx 6.475$