My mathbook tells me that it isn't possible to solve this:
$$2 \sin(x) = \cos(x)$$
But Wolfram Alpha gives the following answer:
$$x = 2\cdot\left(\pi n-\tan^{-1}(2\pm\sqrt{5})\right)$$
Is it possible to do this, without the help of a calculator?
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4$\begingroup$ Well you can find $\tan x$ easily enough the way you've written it - have you got the question right. $\endgroup$– Mark BennetCommented Oct 12, 2015 at 19:34
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2$\begingroup$ The way you wrote it would just be $\tan(x)=\frac 1 2$, so your answer would be $\arctan (\frac 1 2)$ $\endgroup$– AlanCommented Oct 12, 2015 at 19:36
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$\begingroup$ (...together with the solutions obtained by exploiting the symmetry of the tangent function.) $\endgroup$– Travis WillseCommented Oct 12, 2015 at 19:36
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$\begingroup$ Is the question, then, how does one show that WolframAlpha's expression is the solution (or equivalently, equivalent to the solution described here in the comments)? $\endgroup$– Travis WillseCommented Oct 12, 2015 at 19:38
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$\begingroup$ Are both answers correct then? $\endgroup$– AdnanCommented Oct 12, 2015 at 19:44
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1 Answer
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One way can be using $\tan\frac x2=t$ so $\sin x=\frac{2t}{1+t^2}$ and $\cos x=\frac{1-t^2}{1+t^2}$.
Here $2 \sin x= \cos x$ implies $t^2+4t-1=0$ from which $\tan \frac x2=2\pm\sqrt{5}$. Hence the answer of Wolfram.
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$\begingroup$ @Jean Marie: Merci beaucoup professeur. $\endgroup$– PiquitoCommented May 28, 2023 at 14:40