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

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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: wolframalpha.com/input/… Click the "exact form" link for the answers it gives--each one is about one screen-ful of text –  anorton May 3 '13 at 1:39
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@anorton: Solve by hand by computer :-) –  Aryabhata May 3 '13 at 1:41
    
Oh lol I certainly do NOT want to solve for that. –  Ovi May 3 '13 at 1:43

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up vote 3 down vote accepted

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...

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Oh ok. But by the way how did you make this into a solvable quartic equation? –  Ovi May 3 '13 at 1:44
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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 3 '13 at 1:45

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