What is the distance between the following polar coordinates?:  (4,(-7pi)/12), (2,(pi)/8)

Jul 1, 2017

$D \approx 5.4535$

Explanation:

The distance formula for polar coordinates can be derived from the distance formula for rectangular coordinates

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

Instead of using $x$ and $y$ values, though, we would just plug in their polar equivalents

$x = r \cos \left(\theta\right)$

$y = r \sin \left(\theta\right)$

Plugging those and using a couple of trigonometric identities, you get the following in purely polar coordinates

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

Plugging in the polar coordinates you have been given, we get

$D = \sqrt{{\left(4\right)}^{2} + {\left(2\right)}^{2} - 2 \left(4\right) \left(2\right) \cos \left(- \frac{7 \pi}{12} - \frac{\pi}{8}\right)}$

$D = \sqrt{16 + 4 - 16 \cos \left(- \frac{17 \pi}{24}\right)}$

$D \approx \sqrt{20 - 16 \left(- 0.60876\right)}$

$D \approx 5.4535$