# What is the distance between the following polar coordinates?:  (2,(5pi)/12), (1,(3pi)/12)

Jul 18, 2018

$D = \sqrt{5 - 2 \sqrt{3}} \approx 1.2393$

#### Explanation:

We know that ,

$\text{Distance between Polar Co-ordinates:} A \left({r}_{1} , {\theta}_{1}\right) \mathmr{and} B \left({r}_{2} , {\theta}_{2}\right)$ is

color(red)(D=sqrt(r_1^2+r_2^2-2r_1r_2cos(theta_1-theta_2))...to(I)

We have , ${P}_{1} \left(2 , \frac{5 \pi}{12}\right) \mathmr{and} {P}_{2} \left(1 , \frac{3 \pi}{12}\right)$.

So , ${r}_{1} = 2 , {r}_{2} = 1 , {\theta}_{1} = \frac{5 \pi}{12} \mathmr{and} {\theta}_{2} = \frac{3 \pi}{12}$

$\implies {\theta}_{1} - {\theta}_{2} = \frac{5 \pi}{12} - \frac{3 \pi}{12} = \frac{2 \pi}{12} = \frac{\pi}{6} = {30}^{\circ}$

$\implies \cos \left({\theta}_{1} - {\theta}_{2}\right) = \cos \left({30}^{\circ}\right)$

"Using : " color(red)((I) we get

$D = \sqrt{{2}^{2} + {1}^{2} - 2 \left(2\right) \left(1\right) \cos {30}^{\circ}}$

$\implies D = \sqrt{4 + 1 - 4 \cdot \frac{\sqrt{3}}{2}}$

$\implies D = \sqrt{5 - 2 \sqrt{3}}$

$\implies D \approx 1.2393$