Trigonometric equation

Posted November 2nd, 2009 by alpha

Solve for $\theta$ :
$tan\theta+tan2\theta+tan3\theta=0$

Comments

On November 2nd, 2009 Structure says:

$t=\tan \theta$

$\tan 2\theta=\frac{2t}{1-t^2}$

$\tan 3\theta=\frac{3t-t^3}{1-3t^2}$

$t+\frac{2t}{1-t^2}+\frac{3t-t^3}{1-3t^2}$

$\frac{(3-t^2)(2-4t^2)}{(1-t^2)(1-3t^2)}=0$

$t=\tan \theta\ne\pm 1$

$t=\tan \theta\ne\pm \frac{\sqrt 3}{3}$

$t=\tan \theta=\pm\sqrt3$

$t=\tan \theta=\pm\frac{\sqrt 2}{2}$

On November 3rd, 2009 alpha says:

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``Old theorems never die; they turn into definitions.''
----E. Hewitt

''ALPHA" =)