# What is the distance between the following polar coordinates?:  (2,(pi)/4), (5,(5pi)/8)

Apr 16, 2017

$4.6$ to 2dp

#### Explanation:

If we denote $\left(2 , \frac{\pi}{4}\right)$ by $A$, $\left(5 , \frac{5 \pi}{8}\right)$ by $B$ and the origin by $O$, and the angle $\angle A O B$ by $\theta$, Then:

$\theta = \frac{5 \pi}{8} - \frac{\pi}{4} = \frac{3 \pi}{8}$

If we apply the cosine rule:

${a}^{2} = {b}^{2} + {c}^{2} - 2 a b \cos A$

Then we get:

${\left(A B\right)}^{2} = {\left(O A\right)}^{2} + {\left(O B\right)}^{2} - 2 \left(O A\right) \left(O B\right) \cos \theta$
$\text{ } = {\left(2\right)}^{2} + {\left(5\right)}^{2} - 2 \left(2\right) \left(5\right) \cos \left(\frac{3 \pi}{8}\right)$
$\text{ } = 4 + 25 - 20 \cos \left(\frac{3 \pi}{8}\right)$
$\text{ } = 29 - 20 \cos \left(\frac{3 \pi}{8}\right)$
$\text{ } = 21.34633 \ldots$

$\therefore A B = 4.629209$