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$x - 2\arctan(x)= 0$.

I can find one root (0) from the equation

$\tan(x/2) = x$

but there are two others, namely

($-2.3312, 2.3312$)

that I don't know how to find. Looking for help! Thanks :)

link to WolframAlpha solution

David Mitra
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Anton Popov
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  • if WolframAlpha couldn't find an analytical (exact) solution, there probably isn't one. Who's asking? How are you expected to find them? By the way, try "\tan" and "\arctan" for $\tan$ and $\arctan$ – Stefan Smith Nov 14 '13 at 00:22
  • First year calculus course. This is a part of a functional analysis excercise, where $f(x) = x-2\arctan(x)$. I don't really know how I am expected to find them, using math i guess (tee-hee). – Anton Popov Nov 14 '13 at 00:34
  • Did you attempt to find $f'(x)$, the derivative? – abiessu Nov 14 '13 at 00:39
  • Yes, the rest of the problem is fairly easy. My classmates have been finding the answer with plotting calculators or via wolframAlpha (or such). Thought there was an analytical way to solve the roots of the equation vs numerical. Thanks for having a look! – Anton Popov Nov 14 '13 at 00:43
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    @AntonPopov : if this is part of a "functional analysis exercise", then why are the tags (algebra-precalculus) and (trigonometry)? What exercise exactly is this part of? – Stefan Smith Nov 14 '13 at 01:13
  • You can rearrange into $\frac{\tan(y)}y=2$ and use this series – Тyma Gaidash Jun 04 '23 at 14:39

1 Answers1

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Surely you must use a numerical approximation method.
In a calculus book usually you find the bisection method after introducing the intermediate value theorem and the Newton-Raphson method in the chapter on derivatives (near you find also the secant method).
Choose one and you are done (with the comfort of a calculator).

A plot is useful to orient: for example, in this case the roots can be found as abscissas of the points of intersection of the curve $y=\arctan x$ and the line $y=x/2$.

Tony Piccolo
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