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Possible Duplicate:
How to find $x$ in some trigonometric equations

How to solve these trigonometric equations?

$\tan2x-\sin4x = 0$

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    My hint and answer to the last question (as linked by Américo Tavares) give a method of solving this.2012-08-17

2 Answers 2

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HINT: $\sin4x=2\sin2x\cos2x$, and $\tan2x=\dfrac{\sin2x}{\cos2x}$. Now let $a=\sin2x$ and $b=\cos2x$, write your equation in terms of $a$ and $b$, and see what it tells you about $a$ and $b$.

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Since this is a homework, some intermediate steps are omitted and left for you to work out.

The identities you need are$^{\dagger}$ $ \sin(4x) = \color{red}{2} \sin(2x) \cos(2x)\\ \tan(2x) = \frac{\sin(2x)}{\cos(2x)} $ Substitute both in $ \tan(2x)-\sin(4x) = 0$ to get $ (1 - 2 \cos^2(2x))\sin(2x) = 0 $ which you should be able to factor into $3$ cases. Solve each case for $x.$


$^{\dagger}$ Fixed error thanks to Thomas Andrews and David Mitra.

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    I was wondering - did you choose your username after the character from the TV series [Scrubs](http://en.wikipedia.org/wiki/Scrubs_%28TV_series%29)?2012-08-17