# What is the distance between the following polar coordinates?:  (4,pi), (5,pi)

Dec 26, 2015

$1$

#### 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 $\left(4 , \pi\right)$ and $\left({r}_{2} , {\theta}_{2}\right)$ represent $\left(5 , \pi\right)$.
implies d=sqrt(4^2+5^2-2*4*5Cos(pi-pi)
implies d=sqrt(16+25-40Cos(0)
$\implies d = \sqrt{41 - 40 \cdot 1} = \sqrt{41 - 40} = \sqrt{1} = 1$
$\implies d = 1$
Hence the distance between the given points is $1$.

Dec 29, 2015

$1$

#### Explanation:

(this is an attempt to restore my original answer)

Using common insight rather than applying the Pythagorean Theorem and $\cos$ conversions:

The distance between any two polar coordinates with the same angle is the difference in their radii.