Question

Given points (r1, u1 ) and (r2, u2) in polar coordinates, obtain a general formula for the distance between them. Simplify it as much as possible using the identity cos2 u 1 sin2 u 5 1. Hint: Write the expressions for the two points in Cartesian coordinates and substitute into the usual distance formula.

Answer 1

`d=√(r_1^2+r_2^2-2r_1r_2\cos(u_2-u_1))`

Explanation:

The Law of Cosines gives an immediate result. No translation to Cartesian coordinates is necessary. That law makes use of the angle between the vectors, u2-u1

`d^2=r_1^2+r_2^2-2r_1r_2\cos(u_2-u_1)\\\\\boxed{d=√(r_1^2+r_2^2-2r_1r_2\cos(u_2-u_1))}`