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

What is the distance between the following polar coordinates?:
$\left(2,\frac{7\pi }{4}\right),\left(7,\frac{7\pi }{8}\right)$
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Step 1
The two points and the origin form a triangle with sides, $a=2,b=7,$,\ b=7, and the angle between them $C=\frac{7\pi }{4}-\frac{7\pi }{8}=\frac{7\pi }{8}$. Therefore, the distance between the two points will be the length of side, c, and we can use the Law of Cosines to find its length:
$c=\sqrt{{a}^{2}+{b}^{2}-2\left(a\right)\left(b\right)\mathrm{cos}\left(C\right)}$
$c=\sqrt{{2}^{2}+{7}^{2}-2\left(2\right)\left(7\right)\mathrm{cos}\left(\frac{7\pi }{8}\right)}$
$c\approx 8.88$