# What is the distance between the following polar coordinates?:  (7,(23pi)/12), (9,(11pi)/8)

Feb 20, 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:

$d = \sqrt{{7}^{2} + {9}^{2} - \left(2 \times 7 \times 9 \times \cos \left(\frac{23 \pi}{12} - \frac{11 \pi}{8}\right)\right)}$

$d = \sqrt{49 + 81 - \left(126 \times \cos \left(\frac{46 \pi}{24} - \frac{33 \pi}{24}\right)\right)}$

$d \cong \sqrt{130 - \left(126 \times - 0.13\right)}$

$d \cong \sqrt{130 - \left(- 16.38\right)}$

$d \cong \sqrt{130 + 16.38}$

$d \cong \sqrt{146.38}$

$d \cong 12.10$