# What is the distance between the following polar coordinates?:  (2,(10pi)/3), (14,(-31pi)/8)

Nov 16, 2017

$14.39 \textcolor{w h i t e}{88}$units

#### Explanation:

First we need to convert the polar coordinates to Cartesian coordinates. We can do this by using the following:

$x = r \cos \left(\theta\right)$

$y = r \sin \left(\theta\right)$

$\therefore$

$x = 2 \cos \left(\frac{10 \pi}{3}\right) = - 1$

$y = 2 \sin \left(\frac{10 \pi}{3}\right) = - \sqrt{3}$

Cartesian coordinate:

$\left(- 1 , - \sqrt{3}\right)$

$x = 14 \cos \left(\frac{- 31 \pi}{8}\right) = 12.93$

$y = 14 \sin \left(\frac{- 31 \pi}{8}\right) = - 5.36$

Cartesian coordinate:

$\left(12.93 , - 5.36\right)$

Next we use the distance formula:

$d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$

$d = \sqrt{{\left(- 1 - 12.93\right)}^{2} + {\left(- \sqrt{3} + 5.36\right)}^{2}} = 14.39 \textcolor{w h i t e}{88}$units

All results to 2 .d.p.