# What is the distance between (-3 ,(5 pi)/6 ) and (5 , (11 pi )/12 )?

May 31, 2017

$\approx 7.936 \text{ to 3 dec. places}$

#### Explanation:

$\text{the distance ( d ) between 2 "color(blue)"polar coordinates}$ is.

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{{d}^{2} = {r}_{1}^{2} + {r}_{2}^{2} - 2 {r}_{1} {r}_{2} \cos \left({\theta}_{2} - {\theta}_{1}\right)} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\text{where " (r_1.theta_1),(r_2,theta_2)" are 2 points}$

$\text{the points are }$

$\left({r}_{1} , {\theta}_{1}\right) = \left(- 3 , \frac{5 \pi}{6}\right) , \left({r}_{2} , {\theta}_{2}\right) = \left(5 , \frac{11 \pi}{12}\right)$

$\Rightarrow {d}^{2} = {\left(- 3\right)}^{2} + {5}^{2} - \left(2 \times - 3 \times 5 \cos \left(\frac{11 \pi}{12} - \frac{5 \pi}{6}\right)\right)$

$= 9 + 25 + 30 \cos \left(\frac{\pi}{12}\right)$

$\approx 62.9777 . .$

$\Rightarrow d = \sqrt{62.9777 . .} \approx 7.936 \text{ to 3 decimal places}$