# What is the distance between the following polar coordinates?:  (2,(pi)/8), (7,(3pi)/8)

Dec 9, 2017

$\sqrt{53 - 14 \sqrt{2}}$

#### Explanation:

As scene of the picture, i have drawn each of the polar coordinates, with there respective lengths and angles

We can see that $\alpha = \frac{3 \pi}{8} - \frac{\pi}{8} = \frac{\pi}{4}$

Now we can use the cosine rule to find $x$ our distance:

${A}^{2} = {B}^{2} + {C}^{2} - 2 B C \cos a$

$\implies {x}^{2} = {7}^{2} + {2}^{2} - \left(2 \cdot 7 \cdot 2 \cdot \cos \left(\frac{\pi}{4}\right)\right)$

$\implies {x}^{2} = 53 - 14 \sqrt{2}$

As $\cos \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$

Hence yielding:

$\sqrt{53 - 14 \sqrt{2}}$