# What is the distance between (3 ,( pi)/12 ) and (-2 , ( 3 pi )/2 )?

Dec 27, 2016

$4 \sin {7.5}^{o} = \left(\frac{1}{2} \sqrt{2 - \sqrt{2 + \sqrt{3}}}\right)$

$= 4 \left(0.13053\right) = 0.5221$, nearly.

#### Explanation:

I make r positive in the second point and read it as

$\left(2 , \frac{3}{2} \pi - \pi\right) = \left(2 , \frac{\pi}{2}\right)$

The distance PQ between $P \left(a , \alpha\right) \mathmr{and} Q \left(b , \beta\right)$, keeping a

and b non-negative, is

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

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

$= \sqrt{8 \left(1 - \cos \left(\frac{5}{12} \pi\right)\right)}$

$= \sqrt{16 {\sin}^{2} \left(\frac{\pi}{24}\right)}$

$= 4 \sin \left(\frac{\pi}{24}\right)$

$= 4 \sin {7.5}^{o}$

$= 4 \left(\frac{1}{2} \sqrt{2 - \sqrt{2 + \sqrt{3}}}\right)$

$= 4 \left(0.13053\right) = 0.5221$, nearly.2