# Thread: Trigonometric Equation

1. ## Trigonometric Equation

Find the general solution:
tanA+tan2A+tan3A=0

2. Using double and triple angle formulas you can say

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

$\displaystyle \tan(A)+ \frac{2\tan(A)}{1-\tan(A)} +\frac{3\tan(A)-\tan^3(A)}{1-3\tan^2(A)}=0$

Make a common denominator, multiply it through both sides and solve.

3. ## if A = 0

also by observation if $\displaystyle A=0$ than

$\displaystyle tan(0) + tan((2)(0)) + tan((3)(0)) = 0$

$\displaystyle 0+0+0=0$

well one answer anyway

4. Originally Posted by pickslides
Using double and triple angle formulas you can say

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

$\displaystyle \tan(A)+ \frac{2\tan(A)}{1-\tan(A)} +\frac{3\tan(A)-\tan^3(A)}{1-3\tan^2(A)}=0$

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

$\displaystyle \begin{gathered} \tan A + \tan 2A + \tan 3A = 0; \hfill \\ \frac{{\sin 3A}} {{\cos A\cos 2A}} + \frac{{\sin 3A}} {{\cos 3A}} = 0; \hfill \\ \end{gathered}$

$\displaystyle \begin{gathered} \frac{{\sin 3A\cos 3A + \sin 3A\cos A\cos 2A}} {{\cos A\cos 2A\cos 3A}} = 0; \hfill \\ \sin 3A\left( {\cos 3A + \cos A\cos 2A} \right) = 0. \hfill \\ \end{gathered}$

$\displaystyle 1)~~\sin 3A = 0.$

$\displaystyle \begin{gathered} 2)~~\cos 3A + \cos A\cos 2A = 0; \hfill \\ \cos 3A + \frac{1} {2}\left( {\cos A + \cos 3A} \right) = 0; \hfill \\ 3\cos 3A + \cos A = 0; \hfill \\ \end{gathered}$

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

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