How do I find the distance between the points with polar coordinates (3, 120^circ) and (0.5, 49^circ)?

1 Answer
Mar 3, 2015

The distance is 2.88.

As you can see from this drawing:

GeogebraGeogebra

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.

So, in our case:

AB=sqrt(3^2+(1/2)^2-2*3*1/2*cos(120°-49°))=

=sqrt(9+1/4-3cos(71))~=2,88.