# What is the distance between (3 , (3 pi)/8 ) and (9, pi )?

Mar 28, 2016

$\sqrt{90 - 54 \cos \left(5 \frac{\pi}{8}\right)}$ = $\sqrt{110.665} = 10.52$ nearly.

#### Explanation:

The position vectors to the points are of lengths a = 3 and b = 9. The angle in-between is C = $5 \frac{\pi}{8}$.
Use the formula c = $\sqrt{{a}^{2} + {b}^{2} - 2 a b \cos C}$

Mar 28, 2016

$\sqrt{90 + 54 \cos \left(\frac{3 \pi}{8}\right)} \approx 10.520$

#### Explanation:

To convert the polar coordinates to Cartesian coordinates, we use

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

The cartesian coordinate of $\left(3 , \frac{3 \pi}{8}\right)$ is $\left(\frac{3}{2} \sqrt{2 - \sqrt{2}} , \frac{3}{2} \sqrt{2 + \sqrt{2}}\right)$. Use the half angle formula to get the values.

The cartesian coordinate of $\left(9 , \pi\right)$ is $\left(- 9 , 0\right)$.

We can use the Pythagoras Theorem to find the distance between the 2 points

$d = \sqrt{{\left(- 9 - 3 \cos \left(\frac{3 \pi}{8}\right)\right)}^{2} + {\left(0 - 3 \sin \left(\frac{3 \pi}{8}\right)\right)}^{2}}$

$= \sqrt{\left(81 + 9 {\cos}^{2} \left(\frac{3 \pi}{8}\right) + 54 \cos \left(\frac{3 \pi}{8}\right)\right) + 9 {\sin}^{2} \left(\frac{3 \pi}{8}\right)}$

$= \sqrt{90 + 54 \cos \left(\frac{3 \pi}{8}\right)}$

$= 3 \sqrt{3 \sqrt{2 - \sqrt{2}} + 10}$

$\approx 10.520$