# What is the distance between the following polar coordinates?:  (2,(12pi)/8), (7,(7pi)/8)

Jan 7, 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:

$d = \sqrt{{2}^{2} + {7}^{2} - \left(2 \times 2 \times 7 \cos \left(\frac{12 \pi}{8} - \frac{7 \pi}{8}\right)\right)}$

$d = \sqrt{4 + 49 - \left(28 \cos \left(\left(\frac{12}{8} - \frac{7}{8}\right) \pi\right)\right)}$

$d = \sqrt{53 - \left(28 \times - 0.383\right)}$ Rounded to the nearest Thousandth.

$d = \sqrt{53 - \left(- 10.724\right)}$

$d = \sqrt{53 + 10.724}$

$d = \sqrt{63.724}$

$d = 7.983$ Rounded to the Nearest Thousandth.

Or the distance is approximately 8 units.