1. ## Trigonometric Equation

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

2. Using double and triple angle formulas you can say

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

$
\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 $A=0$ than

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

$0+0+0=0$

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

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

$
\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:

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

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

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

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

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

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