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Math Help - Precal, proof for distance between 2 polar points

  1. #1
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    Post 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)
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  2. #2
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    Hello, rwc123!

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


    Using the Law of Cosines

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

    . . is given by:. . 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: . \begin{array}{ccccccc}<br />
|\overrightarrow{OA}| &=& r_1   \\ \\[-4mm]<br />
|\overrightarrow{OB}| &=& r_2 \\ \\[-4mm]<br />
\angle O &=& \theta_2-\theta_1\end{array}


    Law of Cosines:

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


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

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