Finding solution of a Trigonometric equation \tan A+\tan 2A+\tan 3A=0 I tried

Question

Finding solution of a Trigonometric equation \tan A+\tan 2A+\tan 3A=0 I tried

Gregory Jones 2021-12-28 Answered

Finding solution of a Trigonometric equation

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

I tried converting these all in sin and cos and I got the answer but the answer didn't match

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Hattie Schaeffer

Answered 2021-12-29 Author has 37 answers

Use this:

\tan 3 A = \frac{\tan A + \tan 2 A}{1 - \tan A \tan 2 A}

Substitute the numerator with −

\tan 3 A

and youre

This is helpful 0

enhebrevz

Answered 2021-12-30 Author has 25 answers

$\tan A + \tan 2 A = \frac{\sin 3 A}{\cos A \cos 2 A}$
$\tan A + \tan 2 A + \tan 3 A = 0 \Rightarrow \frac{\sin 3 A}{\cos A \cos 2 A} + \frac{\sin 3 A}{\cos 3 A} = 0$
$\Leftrightarrow \sin 3 A (\cos 3 A + \cos A \cos 2 A) = 0$
If $\sin 3 A = 0, 3 A = n π$ where n is any integer
Otherwise,
$\cos 3 A + \cos A \cos 2 A = 0 \Leftrightarrow 4 \cos 3 A - 3 \cos A + \cos A (2 \cos^{2} A - 1) = 0$
$\Leftrightarrow \cos A (6 \cos^{2} A - 4) = 0$
If $\cos A = 0, A = (2 m + 1) \frac{π}{2}$
Otherwise, $6 \cos^{2} A - 4 = 0 \Leftrightarrow \cos^{2} A = \frac{23}{,} \cos 2 A = 2 \cos^{2} A - 1 = \frac{13}{}$

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nick1337

Answered 2022-01-08 Author has 510 answers

Hint
The following identities for the tangent of multiple angles should be useful
$\tan 2 A = \frac{2 \tan A}{1 - 2 \tan^{2} A}$
$\tan 3 A = \frac{3 \tan A - \tan^{3} A}{1 - 3 \tan^{2} A}$
I am sure that you can take from here.