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

What is the distance between the following polar coordinates?: $\left(1,\frac{\pi }{4}\right),\left(12,\frac{9\pi }{8}\right)$
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Step 1
Distance between two polar coordinates formula
$d=\sqrt{{r}_{1}^{2}+{r}_{2}^{2}-2\cdot {r}_{1}\cdot {r}_{2}\cdot \mathrm{cos}\left({\theta }_{2}-{\theta }_{1}\right)}$
Solution:
Let $\left({r}_{1},{\theta }_{1}\right)=\left(1,\frac{\pi }{4}\right)$
Let $\left({r}_{2},{\theta }_{2}\right)=\left(12,\frac{9\pi }{8}\right)$
Use
$d=\sqrt{{r}_{1}^{2}+{r}_{2}^{2}-2\cdot {r}_{1}\cdot {r}_{2}\cdot \mathrm{cos}\left({\theta }_{2}-{\theta }_{1}\right)}$
$d=\sqrt{{1}^{2}+{12}^{2}-2\cdot \left(1\right)\cdot \left(12\right)\cdot \mathrm{cos}\left(\frac{9\pi }{8}-\frac{\pi }{4}\right)}$
$d=\sqrt{145-24\cdot \mathrm{cos}\left(\frac{7\pi }{8}\right)}$
$d=12.92954402832\phantom{\rule{1ex}{0ex}}\text{}\phantom{\rule{1ex}{0ex}}$ units
God bless. I hope the explanation is useful.