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I know I'm supposed to do $\sin(3x) - \sin x = 0$ but beyond that I'm stuck.. I tried expanding $\sin(3x)$ but that didn't help.

  • I want the value of $x$ in the interval $[0, 2\pi)$

2 Answers 2

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We have $\sin{3x} = 3\sin{x} - 4\sin^{3}{x}$ which says that we have to solve the equation $$3\sin{x} - 4\sin^{3}{x} - \sin{x}=0$$, that is $2 \sin{x} - 4\sin^{3}{x}=0$. Take $y = \sin{x}$ and so you have $$2y-4y^{3}=0 \Longrightarrow 2y(1-2y^{2})=0$$ and then see what happens. I hope this helps you out.

Or you can even try this $$\sin{3x} - \sin{x} = 2 \cos\Bigl(\frac{3x+x}{2}\Bigr) \cdot \sin\Bigl(\frac{3x-x}{2}\Bigr) = 2\cos{2x} \cdot \sin{x}$$

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    ow silly me i thought sin3x = 3sinx - 4cos^3(x)2010-09-30
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    Can i divide by y ? and get 2 - 4y^2 ?2010-09-30
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    @Andrei: You should think on your own, from now!2010-09-30
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    Yes thank you! i try to think on my on from now sorry >.<2010-09-30
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    @Andrei: You don't need to be sorry. Just try to think!2010-09-30
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    @andrei: regarding dividing by y, consider whether there any numbers that you cannot divide by.2010-09-30
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    @Isaac i was thinking of for example y < 0 that would affect the equation wouldn't it ?2010-10-01
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    @andrei: If you were dealing with an inequality, $ayb$, but dividing by a negative number is allowed and does not affect an equation. There is, however, a particular single number that you cannot ever divide by—that is, you couldn't even divide 2 by this particular number. What number is this particular number? (I'm trying to give you hints to lead you to the answer, but if this doesn't get you there, let me know and I'll tell you in my next reply.)2010-10-01
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    hm .. is it 0 ?2010-10-16
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You could use the fact that $\sin x=\sin y$ if and only if either $x-y$ is an even integer times $\pi$ or $x+y$ is an odd integer times $\pi$.