What is the distance between the following polar coordinates?:  (3,(-7pi)/12), (7,(5pi)/8)

Jan 6, 2016

$9.556$ units.

Explanation:

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 $\left(3 , \frac{- 7 \pi}{12}\right)$ and $\left({r}_{2} , {\theta}_{2}\right)$ represent $\left(7 , \frac{5 \pi}{8}\right)$.
implies d=sqrt(3^2+7^2-2*3*7Cos((-7pi)/12-(5pi)/8)
implies d=sqrt(9+49-42Cos((-14pi-15pi)/24)
$\implies d = \sqrt{58 - 42 C o s \left(\frac{- 29 \pi}{24}\right)} = \sqrt{58 - 42 \cdot \left(- 0.7933\right)} = \sqrt{58 + 33.3186} = \sqrt{91.3186} = 9.556$ units
$\implies d = 9.556$ units (approx)
Hence the distance between the given points is $9.556$ units.