Question

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

Answer 1

`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 `(r_1,theta_1)` represent `(4,pi)` and `(r_2,theta_2)` represent `(5,pi)`.
`implies d=sqrt(4^2+5^2-2*4*5Cos(pi-pi)`
`implies d=sqrt(16+25-40Cos(0)`
`implies d=sqrt(41-40*1)=sqrt(41-40)=sqrt(1)=1`
`implies d=1`
Hence the distance between the given points is `1`.

Answer 2

`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.