# How do you find the distance between the points with the given polar coordinates P_1(1, pi/6) and P_2(5, (3pi)/4)?

Jul 8, 2018

$\implies D \approx 5.34679$

#### 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 , ${P}_{1} \left(1 , \frac{\pi}{6}\right) \mathmr{and} {P}_{2} \left(5 , \frac{3 \pi}{4}\right)$.

So , ${r}_{1} = 1 , {r}_{2} = 5 , {\theta}_{1} = \frac{\pi}{6} \mathmr{and} {\theta}_{2} = \frac{3 \pi}{4}$

$\implies {\theta}_{1} - {\theta}_{2} = \frac{\pi}{6} - \frac{3 \pi}{4} = \frac{2 \pi - 9 \pi}{12} = - \frac{7 \pi}{12} = - {105}^{\circ}$

$\implies \cos \left({\theta}_{1} - {\theta}_{2}\right) = \cos \left(- {105}^{\circ}\right)$

$\implies \cos \left({\theta}_{1} - {\theta}_{2}\right) = \cos \left({105}^{\circ}\right) \to \left[\because \cos \left(- \theta\right) = \cos \theta\right)$

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

$D = \sqrt{{1}^{2} + {5}^{2} - 2 \left(1\right) \left(5\right) \cos {105}^{\circ}}$

$\implies D = \sqrt{1 + 25 - 10 \cdot \cos {105}^{\circ}}$

$\implies D = \sqrt{26 - 10 \cos {105}^{\circ}}$

$\implies D \approx 5.34679$