Question

What is the distance between `(2 , (5 pi)/8 )` and `(3 , (1 pi )/3 )`?

Answer 1

The distance between those two coordinates is `sqrt(13 - 12 cos((7pi)/24)) ~~2.39`.

Explanation:

You can use the law of cosines to do that.

Let me illustrate why:

Polar coordinates `(r, theta)` are defined by the radius `r` and the angle `theta`.

Imagine lines leading from the pole to your respective polar coordinates. Those lines represent two sides of a triangle with lengths `A = 3` and `B = 2`. The distance between those two coordinates being the third side, `C`.

Furthermore, the angle between `A` and `B` can be computed as the difference between the two angles of your polar coordinates:

`gamma = (5pi)/8 - pi/3 = (7pi)/24`

Thus, the length of the side `C` can be found with the help of law of cosines on that triangle:

`C^2 = A^2 + B^2 - 2AB cos(gamma)`

`= 3^2 + 2^2 - 2 * 3 * 2 * cos((7pi)/24)`

`= 13 - 12 cos((7pi)/24)`

`=> C = sqrt(13 - 12 cos((7pi)/24)) ~~2.39`