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Is it possible to solve (not approximate) the following trigonometric equation by hand? $$\sin(x)+2\sin(x)\cos(x)=\pi/4.$$

Sigur
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Ovi
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    It can be a good idea to plug these sorts of questions into Wolfram Alpha, just to get a feel for the answer to which you're headed. For example: http://www.wolframalpha.com/input/?i=sinx%20%2B%20sin2x%20%3D%20pi%2F4&t=ff3tb01 Click the "exact form" link for the answers it gives--each one is about one screen-ful of text – apnorton May 03 '13 at 01:39
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    @anorton: Solve by hand by computer :-) – Aryabhata May 03 '13 at 01:41
  • Oh lol I certainly do NOT want to solve for that. – Ovi May 03 '13 at 01:43

1 Answers1

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If you are willing to use $\arcsin$ (or $\sin^{-1}$), then yes.

This can be rewritten as a quartic in $\sin x$ which is theoretically solveable by hand (thought it might be very tedious).

Now you take the appropriate roots of those...

Aryabhata
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  • Oh ok. But by the way how did you make this into a solvable quartic equation? – Ovi May 03 '13 at 01:44
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    @Ovi: $2 \sin x \cos x = \frac{\pi}{4} - \sin x$ and square it and use $\cos^2 x = 1 - \sin^2 x$. Actually, that is only one quartic! – Aryabhata May 03 '13 at 01:45