# What is the distance between the following polar coordinates?:  (2,(4pi)/3), (3,(7pi)/6)

##### 1 Answer
Jan 14, 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:

$d = \sqrt{{2}^{2} + {3}^{2} - \left(2 \times 2 \times 3\right) \cos \left(\frac{4 \pi}{3} - \frac{7 \pi}{6}\right)}$

$d = \sqrt{4 + 9 - 12 \cos \left(\left(\frac{2}{2} \times \frac{4 \pi}{3}\right) - \frac{7 \pi}{6}\right)}$

$d = \sqrt{4 + 9 - 12 \cos \left(\frac{8 \pi}{6} - \frac{7 \pi}{6}\right)}$

$d = \sqrt{4 + 9 - 12 \cos \left(\frac{8 \pi - 7 \pi}{6}\right)}$

d = sqrt(4 + 9 - 12cos(pi/6)

$d = \sqrt{13 - 10.392}$

$d = \sqrt{13 - 10.392}$

$d = \sqrt{2.608}$

$d = 1.615$ rounded to the nearest thousandth.