# What is the distance between the following polar coordinates?:  (3,(7pi)/12), (1,(17pi)/8)

Jun 7, 2017

$D \approx 3.04$

#### Explanation:

The distance formula for rectangular coordinates is

$D = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$

Plugging in ${x}_{n} = {r}_{n} \cos \left({\theta}_{n}\right)$ and ${y}_{n} = {r}_{n} \sin \left({\theta}_{n}\right)$, expanding, and simplifying gives

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

Plugging in $\left({r}_{1} , {\theta}_{1}\right) = \left(3 , \frac{7 \pi}{12}\right)$ and $\left({r}_{2} , {\theta}_{2}\right) = \left(1 , \frac{17 \pi}{8}\right)$

$D = \sqrt{{3}^{2} + {1}^{1} - 2 \left(3\right) \left(1\right) \cos \left(\frac{17 \pi}{8} - \frac{7 \pi}{12}\right)}$

$D = \sqrt{10 - 6 \cos \left(\frac{37 \pi}{24}\right)}$

$D \approx 3.04$