# What is the distance between the following polar coordinates?:  (1,pi), (4,pi)

Jan 4, 2018

See a solution process below:

#### Explanation:

The formula for the distance between two polar coordinates is:

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

Where the two points are $\left({r}_{1} , {\Theta}_{1}\right)$ and $\left({r}_{2} , {\Theta}_{2}\right)$

Substituting the values from the points in the problem gives:

$d = \sqrt{{1}^{2} + {4}^{2} - \left(2 \times 1 \times 4\right) \cos \left(\pi - \pi\right)}$

$d = \sqrt{1 + 16 - 8 \cos \left(0\right)}$

$d = \sqrt{17 - \left(8 \times 1\right)}$

$d = \sqrt{17 - 8}$

$d = \sqrt{9}$

$d = 3$

The distance between the two points is 3 units.