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

Mar 3, 2015

The answer is: $\cong 5 , 4$.

As you can see from this drawing:

$A$ and $B$ in polar coordinates are $A \left(O A , \alpha\right)$ and $B \left(O B , \beta\right)$.

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 $\cos \gamma = \cos \left(- \gamma\right)$).

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

So we can use the cosine theorem, that says:

${a}^{2} = {b}^{2} + {c}^{2} - 2 b c \cos \alpha$,

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.