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Math Help - Converting irrational number to surd form

  1. #1
    Senior Member
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    Converting irrational number to surd form

    I have the following function:

    y = \frac{cosx}{1 - sinx} where x = \frac{\pi}{6}

    Say I solve this for y, and I get the solution 1.73205

    I happen to know this is \sqrt{3}, but if I didn't, is there an intuitive way to make this connection? It seems to be bad form to express it as a decimal.

    Thanks.
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  2. #2
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    Lexington, MA (USA)
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    Hello, Glitch!

    I have the following function:
    . .
    y \:=\: \dfrac{\cos x}{1 - \sin x} where x = \frac{\pi}{6}

    Say I solve this for y, and I get the solution 1.73205

    I happen to know this is \sqrt{3},
    but if I didn't, is there an intuitive way to make this connection?
    It seems to be bad form to express it as a decimal.

    We're expected to know about the 30-60 right triangle.

    Code:
                                  *
                              * 60*
                    2     *       *
                      *           * 1
                  *               *
              * 30                *
          *   *   *   * _ *   *   *
                       √3

    Hence, we have: . \begin{Bmatrix} \sin30^o &=& \sin\frac{\pi}{6} &=& \dfrac{1}{2} \\ \\[-3mm] \cos30^o &=& \cos\frac{\pi}{6} &=&\dfrac{\sqrt{3}}{2} \\ \\[-3mm] \tan30^o &=& \tan\frac{\pi}{6} &=& \dfrac{1}{\sqrt{3}} \end{Bmatrix}


    So we have: . y \;=\;\dfrac{\cos\frac{\pi}{6}}{1-\sin\frac{\pi}{6}} \;=\;\dfrac{\frac{\sqrt{3}}{2}}{1 - \frac{1}{2}} \;=\;\dfrac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} \;=\;\sqrt{3}

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  3. #3
    Senior Member
    Joined
    Apr 2010
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    Ahh, I had forgotten about that triangle! Thanks.
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