# What is the distance between the following polar coordinates?:  (5,(-5pi)/12), (5,(11pi)/6)

Jun 18, 2017

$D = \sqrt{50 + 50 \frac{\sqrt{2}}{2}} \approx 9.2388$

#### Explanation:

We are given

${r}_{1} = 5$
${\theta}_{1} = - \frac{5 \pi}{12}$

${r}_{2} = 5$
${\theta}_{2} = \frac{11 \pi}{6}$

The distance formula for polar coordinates is

$D = \sqrt{{r}_{1}^{2} + {r}_{2}^{2} - 2 {r}_{1} {r}_{2} \cos \left({\theta}_{1} - {\theta}_{2}\right)}$

$D = \sqrt{{5}^{2} + {5}^{2} - 2 \left(5\right) \left(5\right) \cos \left(- \frac{5 \pi}{12} - \frac{11 \pi}{6}\right)}$

$D = \sqrt{25 + 25 - 50 \cos \left(- \frac{9 \pi}{4}\right)}$

$D = \sqrt{50 + 50 \frac{\sqrt{2}}{2}} \approx 9.2388$