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Math Help - Trigonometric Equation

  1. #1
    Junior Member
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    Unhappy Trigonometric Equation

    Find the general solution:
    tanA+tan2A+tan3A=0
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  2. #2
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    Using double and triple angle formulas you can say

    \tan(A)+\tan(2A)+\tan(3A) = 0

     <br />
\tan(A)+ \frac{2\tan(A)}{1-\tan(A)} +\frac{3\tan(A)-\tan^3(A)}{1-3\tan^2(A)}=0<br />

    Make a common denominator, multiply it through both sides and solve.
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  3. #3
    Super Member bigwave's Avatar
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    Cool if A = 0

    also by observation if A=0 than

    tan(0) + tan((2)(0)) + tan((3)(0)) = 0

    0+0+0=0

    well one answer anyway
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  4. #4
    Senior Member DeMath's Avatar
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    Quote Originally Posted by pickslides View Post
    Using double and triple angle formulas you can say

    \tan(A)+\tan(2A)+\tan(3A) = 0

     <br />
\tan(A)+ \frac{2\tan(A)}{1-\tan(A)} +\frac{3\tan(A)-\tan^3(A)}{1-3\tan^2(A)}=0<br />

    Make a common denominator, multiply it through both sides and solve.
    It is better to do so:

    \begin{gathered}<br />
  \tan A + \tan 2A + \tan 3A = 0; \hfill \\<br />
  \frac{{\sin 3A}}<br />
{{\cos A\cos 2A}} + \frac{{\sin 3A}}<br />
{{\cos 3A}} = 0; \hfill \\ <br />
\end{gathered}

    \begin{gathered}<br />
  \frac{{\sin 3A\cos 3A + \sin 3A\cos A\cos 2A}}<br />
{{\cos A\cos 2A\cos 3A}} = 0; \hfill \\<br />
  \sin 3A\left( {\cos 3A + \cos A\cos 2A} \right) = 0. \hfill \\ <br />
\end{gathered}

    1)~~\sin 3A = 0.

    \begin{gathered}<br />
  2)~~\cos 3A + \cos A\cos 2A = 0; \hfill \\<br />
  \cos 3A + \frac{1}<br />
{2}\left( {\cos A + \cos 3A} \right) = 0; \hfill \\<br />
  3\cos 3A + \cos A = 0; \hfill \\ <br />
\end{gathered}

    \begin{gathered}<br />
  3\cos A\left( {4{{\cos }^2}A - 3} \right) + \cos A = 0; \hfill \\<br />
  3{\cos ^3}A - 2\cos A = 0; \hfill \\<br />
  \cos A\left( {\sqrt 3 \cos A - \sqrt 2 } \right)\left( {\sqrt 3 \cos A + \sqrt 2 } \right) = 0. \hfill \\<br />
   \hfill \\ <br />
\end{gathered}
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