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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 $[0, 2\pi)$
Stahl
  • 23,212
andrei
  • 235

2 Answers2

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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 $\pi$.

Robin Chapman
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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 $$\sin{3x} - \sin{x} = 2 \cos\Bigl(\frac{3x+x}{2}\Bigr) \cdot \sin\Bigl(\frac{3x-x}{2}\Bigr) = 2\cos{2x} \cdot \sin{x}$$

  • ow silly me i thought sin3x = 3sinx - 4cos^3(x) – andrei Sep 30 '10 at 19:08
  • Can i divide by y ? and get 2 - 4y^2 ? – andrei Sep 30 '10 at 19:10
  • @Andrei: You should think on your own, from now! –  Sep 30 '10 at 19:26
  • Yes thank you! i try to think on my on from now sorry >.< – andrei Sep 30 '10 at 20:06
  • @Andrei: You don't need to be sorry. Just try to think! –  Sep 30 '10 at 20:07
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    @andrei: regarding dividing by y, consider whether there any numbers that you cannot divide by. – Isaac Sep 30 '10 at 20:27
  • @Isaac i was thinking of for example y < 0 that would affect the equation wouldn't it ? – andrei Oct 01 '10 at 14:06
  • @andrei: If you were dealing with an inequality, $ay<by$, then dividing by $y<0$ would reverse the sense of the inequality to $a>b$, 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.) – Isaac Oct 01 '10 at 19:07
  • hm .. is it 0 ? – andrei Oct 16 '10 at 17:04