# What is the distance between the following polar coordinates?:  (3,(-7pi)/3), (1,(3pi)/4)

Jan 12, 2016

The distance formula for polar coordinates is

d=sqrt(r_1^2+r_2^2-2r_1r_2Cos(theta_1-theta_2)
Where $d$ is the distance between the two points, ${r}_{1}$, and ${\theta}_{1}$ are the polar coordinates of one point and ${r}_{2}$ and ${\theta}_{2}$ are the polar coordinates of another point.
Let $\left({r}_{1} , {\theta}_{1}\right)$ represent $\left(3 , \frac{- 7 \pi}{3}\right)$ and $\left({r}_{2} , {\theta}_{2}\right)$ represent $\left(1 , \frac{3 \pi}{4}\right)$.
implies d=sqrt(3^2+1^2-2*3*1Cos((-7pi)/3-(3pi)/4)
implies d=sqrt(9+1-6Cos((-28pi-9pi)/12)
implies d=sqrt(10-6Cos((-37pi)/12)
$\implies d = \sqrt{10 - 6 \left(- 0.9659\right)}$
$\implies d = \sqrt{10 + 5.7954} = \sqrt{15.7954} = 3.9743$ units
$\implies d = 3.9743$ units (approx)
Hence the distance between the given points is $3.9743$ units.