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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4Well you can find $\tan x$ easily enough the way you've written it - have you got the question right. – Mark Bennet Oct 12 '15 at 19:34
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2The way you wrote it would just be $\tan(x)=\frac 1 2$, so your answer would be $\arctan (\frac 1 2)$ – Alan Oct 12 '15 at 19:36
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(...together with the solutions obtained by exploiting the symmetry of the tangent function.) – Travis Willse Oct 12 '15 at 19:36
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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)? – Travis Willse Oct 12 '15 at 19:38
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Are both answers correct then? – Adnan Oct 12 '15 at 19:44
1 Answers
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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.
Jean Marie
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Ataulfo
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