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I need to solve the following equation: $(-12\cos^2(x)-2\cos(x)+3)(-\sin(x))=0$ on the interval $(0,2\pi)$

I found 3 out of 5 solutions which are the following: $\pi, \arccos\left(\frac{-1-\sqrt{37}}{12}\right) \mbox{ and } \arccos\left(\frac{\sqrt{37}-1}{12}\right)$

Can someone help me find the 2 other solutions

Thank you in advance

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    I see on the graph that there exists two additional solutions2012-10-29

2 Answers 2

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Observe that $\cos x=\cos (2\pi-x)$. So if $\alpha$ is a solution, then so is $2\pi-\alpha$.
So you have $\pi, \arccos\left(\frac{-1-\sqrt{37}}{12}\right), \arccos\left(\frac{-1+\sqrt{37}}{12}\right),2\pi-\arccos\left(\frac{-1-\sqrt{37}}{12}\right),2\pi-\arccos\left(\frac{-1+\sqrt{37}}{12}\right)$
5 solutions, as required.

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    @user43418: You're welcome :-)2012-10-29
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your answer is here,

take a look at:

wolfram alpha gives this answer

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    but copy all of the thing, sorry (I should tell it beforehand)2012-10-29