What's the difference in finding the distance between two polar coordinates and two rectangular coordinate?

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Answer 1 · 2015-03-11T21:17:37

Hello,

$d = sqrt((x-x')^2 + (y-y')^2)$ .

With polar coordinates, $A[t, theta]$ and $A'[r',theta']$ , you have to write the relations :

$x = r cos theta, y = r sin theta$
$x' = r' cos theta', y' = r' sin theta'$ ,

So,

$d = sqrt((r cos theta - r' cos theta')^2 + (r sin theta - r' sin theta')^2 )$

Develop, and use the formula $cos^2 x + sin^2 x = 1$ . So you get :

$d = sqrt(r^2 - 2 rr' (cos theta cos theta' + sin theta sin theta')+ r'^2)$

Finally, you know that $cos theta cos theta' + sin theta sin theta' = cos(theta - theta')$ , therefore,

$d = sqrt(r^2 + r'^2 - 2rr' cos(theta - theta'))$ .