# What is the distance between the following polar coordinates?:  (5,(3pi)/4), (3,(11pi)/8)

Aug 14, 2017

The distance is approximately $6.74$ units. See explanation.

#### Explanation:

If two polar points $A = \left({r}_{1} , {\varphi}_{1}\right)$ and $B = \left({r}_{2} , {\varphi}_{2}\right)$ are given, then the distance between the points can be calculated as:

$d \left(A , B\right) = \sqrt{{r}_{1}^{2} + {r}_{2}^{2} - 2 {r}_{1} {r}_{2} \cos \left({\varphi}_{2} - {\varphi}_{1}\right)}$

Here we have:

$d = \sqrt{{5}^{2} + {3}^{2} - 2 \cdot 5 \cdot 3 \cdot \cos \left(\frac{11 \pi}{8} - \frac{3 \pi}{4}\right)} =$

$= \sqrt{25 + 9 - 30 \cos \left(\frac{11 \pi - 6 \pi}{8}\right)} =$

$= \sqrt{34 - 30 \cos \left(\frac{5 \pi}{8}\right)} \approx \sqrt{34 - 30 \cdot \left(- 0.383\right)} \approx$

$\approx \sqrt{34 + 11.49} \approx \sqrt{45.49} \approx 6.74$