Question

What is the distance between polar coordinates `(-4, -30^circ)` and `(3, 250^circ)`?

Answer 1

The answer is: `~=5,4`.

As you can see from this drawing:

Geogebra

`A` and `B` in polar coordinates are `A(OA,alpha)` and `B(OB,beta)`.

I have named `gamma` the angle between the two vectors `v_1` and `v_2`, and, as you can easily see `gamma=alpha-beta`.
(It's not important if we do `alpha-beta` or `beta-alpha` because, at the end, we will calculate the cosine of `gamma` and `cosgamma=cos(-gamma)`).

We know, of the triangle `AOB`, two sides and the angle between them and we have to find the segment `AB`, that is the distance between `A` and `B`.

So we can use the cosine theorem, that says:

`a^2=b^2+c^2-2bc cosalpha`,

where `a,b,c` are the three sides of a triangle and `alpha` is the angle between `b` and `c`.

In our case:

`AB=sqrt((-4)^2+3^2-2*(-4)*3*cos(-280°))=`

`sqrt(16+9+24cos280°)~=5,4`.