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$$\tan4x=\sqrt 3,\qquad 0\leq x \lt \pi$$

4 solutions: $$ \frac{\pi}{12}, \frac{\pi}{3}, \frac{7\pi}{12}, \frac{5\pi}{6}$$

or 3 solutions: $$ \frac{\pi}{12}, \frac{\pi}{3}, \frac{7\pi}{12}$$ (text suggested this and no clues about why $$\frac{5\pi}{6}$$ was excluded

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    It was excluded because books and "official" solutions sometimes have mistakes. And they should have given all the answers with denominator $12$, simplifying hides structure.2012-09-28
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    verifying textbook now, since it is quite credible, i wonder if i made that wrong2012-09-28

2 Answers 2

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Ah, yes, the original formulation of the question shows why the book only has three answers. If you plug in $\frac{5\pi}{6}$ then it gets multiplied by the $3$ in two of the tangent terms and $$\tan({3(\frac{5\pi}{6})})=\tan{\frac{5\pi}{2}}$$ is not defined.

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    ya, thanks, i got that now2012-09-29
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Edit: forget what I said, it should be 4 solutions, as the comment says, ($20\pi/6$) is ($ < 4\pi$)

I put it in my mind as $25\pi/6$

$\text{let}\; \theta = 4x$

$ 0 < \theta < 4\pi$

If you now solve for $\theta$, you will get four solutions.

Sorry about that....

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    But $20/6 < 4$.2012-09-28
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    I am not following, $$\frac {20\pi}{6} = 3.3333333\pi \lt 4\pi$$?2012-09-28
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    i count 4 if I solve for $$\theta$$? $$4\pi$$ is two revolution, the $$1^{st}$$ quadrant, $$3^{rd}$$ quadrant, $$5^{th}$$ quadrant, $$7^{th}$$ quardrant2012-09-28
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    Sorry about the confusion, answer edited..2012-09-28
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    i don't know if i made compound angles right, the original question is $$tanx+tan3x+\sqrt{3}tanxtan3x=tanxtan3x$$, same range for $$\theta$$, does it justifty 3 answers?2012-09-28
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    Are you sure you copied the original question properly here? because the two functions aren't the same. Try plotting them and see.2012-09-28
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    @PaulSmith: it looks like you're using too many dollar signs. Use double dollar signs only when it's necessary to create a new line. Example: $\$$x$\$$ produces $x$, while $\$\$$x$\$\$$ produces $$x$$.2012-09-28
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    @PaulSmith : If you write \tan x\tan3x, with backslashes, then that gives you proper spacing before and after $\tan$, and causes it not to be italicized, thus: $\tan x\tan3x$.2012-09-28
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    Abdulhaq. Which two functions are not the same?2012-09-29
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    Michael. Thank you for your advice, i will pay attention to how I typed latex now.2012-09-29
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    The one you wrote in the comment (the original question in the textbook) and the one in the main question. Maybe you could post the steps you took to go from the original question to $\tan 4x = \sqrt{3}$2012-09-29