# Precal, proof for distance between 2 polar points

• Nov 21st 2008, 09:44 AM
rwc123
Precal, proof for distance between 2 polar points
Use the Law of Cosines to prove that the distance between 2 polar points (r1, θ1) and (r2, θ2) is d2=r12+r22-2r1r2cos(θ2-θ1)
• Nov 21st 2008, 01:08 PM
Soroban
Hello, rwc123!

I don't see any difficulty . . . am I missing something?

Quote:

Using the Law of Cosines

. . prove that the distance $\displaystyle d$ between 2 polar points $\displaystyle A(r_1,\:\theta_1)\text{ and }B(r_2,\:\theta_2)$

. . is given by:. . $\displaystyle d^2\;=\;r_1^2+ r_2^2 -2(r_1)(r_2)\cos(\theta_2-\theta_1)$

Code:

                            B                             *                           *  *                         *      * d                       *          *                     *              *             r2  *                  * A                 *                *               *  θ2 - θ1    *             *        *    r1           *    *         *  *     O *

We have: .$\displaystyle \begin{array}{ccccccc} |\overrightarrow{OA}| &=& r_1 \\ \\[-4mm] |\overrightarrow{OB}| &=& r_2 \\ \\[-4mm] \angle O &=& \theta_2-\theta_1\end{array}$

Law of Cosines:

. . . $\displaystyle d^2\;=\;|AB|^2 \;=\;|OA|^2 + |OB|^2 - 2(|OA|)(|OB|)\cos(\angle O)$

Therefore: .$\displaystyle d^2 \;=\;r_1^2 + r_2^2 - 2(r_1)(r_2)\cos(\theta_2-\theta_1)$