# What is the distance between the following polar coordinates?:  (3,(17pi)/12), (-8,(5pi)/8)

Jan 15, 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:

$d = \sqrt{{3}^{2} + {\left(- 8\right)}^{2} - \left(2 \cdot 3 \cdot - 8\right) \cos \left(\frac{17 \pi}{12} - \frac{5 \pi}{8}\right)}$

d = sqrt(9 + 64 - (-48)cos((2/2 xx (17pi)/12) - (3/3 xx (5pi)/8))

$d = \sqrt{73 + 48 \cos \left(\frac{34 \pi}{24} - \frac{15 \pi}{24}\right)}$

$d = \sqrt{73 + 48 \cos \left(\frac{34 \pi - 15 \pi}{24}\right)}$

$d = \sqrt{73 + 48 \cos \left(\frac{19 \pi}{24}\right)}$

$d = \sqrt{73 + \left(48 \cdot - 0.793\right)}$

$d = \sqrt{73 + \left(- 38.081\right)}$

$d = \sqrt{34.919}$

$d = 5.909$ rounded to the nearest thousandth.