# What is the distance between (2 ,(7 pi)/4 ) and (5 , (11 pi )/12 )?

Jul 4, 2018

$D = \sqrt{29 + 10 \sqrt{3}}$

$\mathmr{and} D \approx 6.81$

#### Explanation:

We know that ,

$\text{Distance between two polar co-ordinates }$

$A \left({r}_{1} , {\theta}_{1}\right) \mathmr{and} B \left({r}_{2} , {\theta}_{2}\right)$ is :

color(red)(D=sqrt(r_1^2+r_2^2-2r_1r_2cos(theta_1-theta_2))...to(1)

We have ,$A \left(2 , \frac{7 \pi}{4}\right) \mathmr{and} B \left(5 , \frac{11 \pi}{12}\right)$

$\therefore {r}_{1} = 2 , {r}_{2} = 5 , {\theta}_{1} = \frac{7 \pi}{4} \mathmr{and} {\theta}_{2} = \frac{11 \pi}{12}$

$\therefore \cos \left({\theta}_{1} - {\theta}_{2}\right) = \cos \left(\frac{7 \pi}{4} - \frac{11 \pi}{12}\right) = \cos \left(\frac{21 \pi - 11 \pi}{12}\right)$

$\implies \cos \left({\theta}_{1} - {\theta}_{2}\right) = \left(\frac{- 10 \pi}{12}\right) = \cos \left(- \frac{5 \pi}{6}\right)$

$\implies \cos \left({\theta}_{1} - {\theta}_{2}\right) = \cos \left(\frac{5 \pi}{6}\right) \ldots \to \left[\because \cos \left(- \theta\right) = \cos \theta\right]$

$\cos \left({\theta}_{1} - {\theta}_{2}\right) = \cos \left(\pi - \frac{\pi}{6}\right) = - \cos \left(\frac{\pi}{6}\right) \to \left[\because I {I}^{n d} Q u a d .\right]$

$\implies \cos \left({\theta}_{1} - {\theta}_{2}\right) = - \frac{\sqrt{3}}{2}$

So ,from $\left(1\right)$

$D = \sqrt{{2}^{2} + {5}^{2} - 2 \left(2\right) \left(5\right) \left(- \frac{\sqrt{3}}{2}\right)}$

$\implies D = \sqrt{4 + 25 + 20 \left(\frac{\sqrt{3}}{2}\right)}$

$\implies D = \sqrt{29 + 10 \sqrt{3}}$

$\implies D \approx 6.81$