# What is the distance between (-9 ,( 17 )/12 ) and (-2 ,( 5 pi )/4 )?

Nov 30, 2016

see below

#### Explanation:

Each point on a polar plane is represented by the ordered pair $\left(r , \theta\right)$.

So lets call the coordinates of ${P}_{1}$ as $\left({r}_{1} , {\theta}_{1}\right)$ and coordinates of ${P}_{2}$ as $\left({r}_{2} , {\theta}_{2}\right)$ . To find the distance between two points on a polar plane use the formula $d = \sqrt{{\left({r}_{1}\right)}^{2} + {\left({r}_{2}\right)}^{2} - 2 {r}_{1} {r}_{2} \cos \left({\theta}_{2} - {\theta}_{1}\right)}$

Thererfore using the points $\left(- 9 , \frac{17}{12}\right)$ and $\left(- 2 , \frac{5 \pi}{4}\right)$, and the formula

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

we have

$d = \sqrt{{\left(- 9\right)}^{2} + {\left(- 2\right)}^{2} - 2 \cdot - 9 \cdot - 2 \cos \left(\frac{5 \pi}{4} - \frac{17}{12}\right)}$

$\therefore d \approx 10.68$