# What is the distance between the following polar coordinates?:  (5,(-15pi)/12), (5,(-7pi)/8)

May 12, 2017

$5.56$ (2dp)

#### Explanation:

Let $A = \left(5 , \frac{- 15 \pi}{12}\right)$, and $B = \left(5 , \frac{- 7 \pi}{8}\right)$

We can plot the polar coordinates as follows:

The angle between $A$ and $B$ is therefore given by:

$\angle A O B = \frac{- 7 \pi}{8} - \frac{- 15 \pi}{12} = \frac{3}{8} \pi$

By the cosine rule we have:

$A {B}^{2} = O {A}^{2} + O {B}^{2} - 2 \left(O A\right) \left(O B\right) \cos \left(\angle A O B\right)$
$\text{ } = {5}^{2} + {5}^{2} - 2 \left(5\right) \left(5\right) \cos \left(\frac{3}{8} \pi\right)$
$\text{ } = 25 + 25 - 50 \cos \left(\frac{3}{8} \pi\right)$
$\text{ } = 50 - 50 \cos \left(\frac{3}{8} \pi\right)$
$\text{ } = 30.865828 \ldots$

$\therefore A B = 5.555702 \ldots$