Question

How do I find the distance between the points with polar coordinates `(2, 10^circ)` and `(7, 10^circ)`?

Answer 1

The answer is: `5`.

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(2^2+7^2-2*2*7*cos0°)=`

`=sqrt(4+49-28*1)=5`.

N.B.:

This is a very particular case, because the angle is the same for both the points (`10°`), so the two points are on the same line, and their distance could be easily calculate `AB=7-2=5`!