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

What is the distance between the following polar coordinates?: $\left(1,\frac{3\pi }{4}\right),\left(5,\frac{5\pi }{8}\right)$
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Step 1
Call the two points A and B:
$A=\left({r}_{a},{\theta }_{a}\right)=\left(1,\frac{3\pi }{4}\right)$
$B=\left({r}_{b},{\theta }_{b}\right)=\left(5,\frac{5\pi }{8}\right)$
Convert them to rectangular form:
$A=\left({x}_{a},{y}_{a}\right)$
${x}_{a}={r}_{a}\mathrm{cos}\left({\theta }_{a}\right)=1\cdot \mathrm{cos}\left(\frac{3\pi }{4}\right)=-\frac{\sqrt{2}}{2}$
${y}_{a}={r}_{a}\mathrm{sin}\left({\theta }_{a}\right)=1\cdot \mathrm{sin}\left(\frac{3\pi }{4}\right)=\frac{\sqrt{2}}{2}$
$A=\left(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)\approx \left(-0.707,0.707\right)$
$B=\left({x}_{b},{y}_{b}\right)$
${x}_{b}={r}_{b}\mathrm{cos}\left({\theta }_{b}\right)=5\cdot \mathrm{cos}\left(\frac{5\pi }{8}\right)=-5\frac{\sqrt{2-\sqrt{2}}}{2}$
${y}_{b}={r}_{b}\mathrm{sin}\left({\theta }_{b}\right)=5\cdot \mathrm{sin}\left(\frac{5\pi }{8}\right)=5\frac{\sqrt{2+\sqrt{2}}}{2}$
$B=\left(-5\frac{\sqrt{2-\sqrt{2}}}{2},5\frac{\sqrt{2+\sqrt{2}}}{2}\right)\approx \left(-1.913,4.619\right)$
Now apply the Pythagorean Theorem to find the length of the line segment $\overline{AB}$
$||\overline{AB}||=\sqrt{{\left({x}_{b}-{x}_{a}\right)}^{2}+{\left({y}_{b}-{y}_{a}\right)}^{2}}$
$||\overline{AB}||\approx \sqrt{{\left(\left(-1.913\right)-\left(-0.707\right)\right)}^{2}+{\left(4.619-0.707\right)}^{2}}$
$||\overline{AB}||\approx \sqrt{{\left(-1.206\right)}^{2}+{\left(3.912\right)}^{2}}=\sqrt{1.455+15.306}$
$||\overline{AB}||\approx \sqrt{16.761}\approx 4.094$