Trigonometric equation, missing some solutions

I'm missing part of the answer, and I'm not quite sure why. The given answer doesn't even seem to hold...

Solve for x: $$\tan 2x = 3 \tan x$$

First some simplifications:

$$\tan 2x = 3 \tan x$$

$$\tan 2x - 3 \tan x = 0$$

$$\frac{\sin 2x}{\cos 2x} - \frac{3 \sin x}{\cos x} = 0$$

$$\frac{2 \sin x \cos^2x - 3 \sin x \cos 2x}{\cos 2x \cos x} = 0$$

$$\frac{\sin x(2 \cos^2x - 3 \cos 2x)}{\cos 2x \cos x} = 0$$

$$\frac{\sin x(2 \cos^2x - 3 (\cos^2 x - \sin^2 x))}{\cos 2x \cos x} = 0$$

$$\frac{\sin x(2 \cos^2x - 3\cos^2 x + \sin^2 x)}{\cos 2x \cos x} = 0$$

$$\frac{\sin x(\sin^2 x - \cos^2 x)}{\cos 2x \cos x} = 0$$

$$\frac{\sin x(\sin^2 x - \cos^2 x)}{(\sin^2 x - \cos^2 x) \cos x} = 0$$

$$\frac{\sin x}{\cos x} = 0$$

Looks much simpler. Now solving for x, since $\frac{\sin x}{\cos x} = 0$ when $\sin x = 0$ and $\sin x = 0$ for every half rotation, the answer must be $k\pi$.

Alas, according to my answer sheet, I'm missing two values: $\frac{\pi}{6} + k\pi$ and $\frac{5\pi}{6} + k\pi$. But since $\frac{\sin(\frac{\pi}{6})}{\cos(\frac{\pi}{6})} = \frac{\sqrt3}{3}$, I'm not sure where these answers come from.

Furthermore, this is the kind of mistake I'm making all over these exercises, I'd like to avoid that, but how can I be sure I have ALL the answers needed?

• Between 6th and 7th line, you forgot to multiply $\sin^2 x$ by $3$ (from $\frac{\sin x(2\cos^2 x-3(\cos^2x-\sin^2x))}{\cos2x \cos x}$ to $\frac{\sin x(2\cos^2 x-3\cos^2x+\sin^2x)}{\cos2x \cos x}$ ) – Milly Nov 5 '14 at 15:37
• Owh, that's sloppy, so my simplification is entirely wrong! – Apeiron Nov 5 '14 at 15:40

3 Answers

Setting $\tan x=t$

we have $$\frac{2t}{1-t^2}=3t\iff2t=3t(1-t^2)\iff t(2-3+3t^2)=0$$

If $t=0,\tan x=0, x=n\pi$ where $n$ is any integer

$2-3+3t^2=0\iff 3t^2=1\implies\cos2x=\dfrac{1-t^2}{1+t^2}=\dfrac12=\cos\dfrac\pi3$

$\implies2x=2m\pi\pm\dfrac\pi3$ where $m$ is any integer

Hint:

You should use $$\tan 2x=\frac{2\tan x}{1-\tan^2x}$$

By your method, $$\frac{\sin x(2 \cos^2x - 3 \cos 2x)}{\cos 2x \cos x} = 0$$

Either $\sin x=0\implies x=n\pi$ where $n$ is any integer

else $2 \cos^2x - 3 \cos 2x=0\iff 1+\cos2x-3\cos2x=0\iff\cos2x=\dfrac12=\cos\dfrac\pi3$

The rest is like my other answer