# What is the distance between the following polar coordinates?:  (6,(17pi)/12), (5,(9pi)/8)

##### 2 Answers
Apr 27, 2017
1. They have different radii
2. They have different angles

#### Explanation:

Polar coordinates are written in the following format:
$\left(r , \setminus \theta\right)$
Where $r$ is the distance from the origin and $\setminus \theta$ is the angle rotated counterclockwise about the origin from the x-axis. You can convert to polar coordinates by using the following formulas.
$x = r \cos \left(\setminus \theta\right)$
$y = r \sin \left(\setminus \theta\right)$
$\tan \left(\setminus \theta\right) = \setminus \frac{y}{x}$
If you plot your points in either Cartesian or polar coordinates, you can clearly see the difference.

Apr 27, 2017

$\text{Distance } \approx 10.42119$

#### Explanation:

$\left(6 , \frac{17 \pi}{12}\right) , \left(5 , \frac{5 \pi}{8}\right)$

Convert to Cartesian coordinates, remember the following formulas:
$\textcolor{red}{x = r \cos \theta}$
$\textcolor{red}{y = r \sin \theta}$

First coordinate: $\left(6 , \frac{17 \pi}{12}\right)$
$x = 6 \cos \left(\frac{17 \pi}{12}\right)$

$y = 6 \sin \left(\frac{17 \pi}{12}\right)$

color(blue)((6cos((17pi)/12),6sin((17pi)/12))

Second coordinate: $\left(5 , \frac{5 \pi}{8}\right)$
$x = 5 \cos \left(\frac{5 \pi}{8}\right)$

$y = 5 \sin \left(\frac{5 \pi}{8}\right)$

color(blue)((5cos((5pi)/8),5sin((5pi)/8))

Use the distance formula (Pythagorean Theorem) between these points:
$\text{Distance} = \sqrt{{\left(6 \cos \left(\frac{17 \pi}{12}\right) - 5 \cos \left(\frac{5 \pi}{8}\right)\right)}^{2} + {\left(6 \sin \left(\frac{17 \pi}{12}\right) - 5 \sin \left(\frac{5 \pi}{8}\right)\right)}^{2}}$

$\text{Distance } \approx 10.42119$