# $\tan(x) = x$. Find the values of $x$

How can I find the possible values of $x$ for:

$\tan(x)=x$

mathematically?

• $x=0{}{}{}{}{}$. – Daryl Jan 3 '13 at 20:29
• You can get very nice estimates for the roots. Some non-trivial mathematics is involved. There is as far as I know no "closed form" expression for the family of solutions. – André Nicolas Jan 3 '13 at 20:29
• Note that for $-\pi/2 < x < \pi/2$, only $x=0$ is a solution. But there are infinitely many solutions with $x$ outside this region, and, as Andre says, it is unlikely they have a closed form. – Thomas Andrews Jan 3 '13 at 20:31
• Let $k$ be any integer. Then there is exactly one solution $x$ in the interval $(k\pi -\pi/2,k\pi+\pi/2)$ – Thomas Andrews Jan 3 '13 at 20:39
• Degrees, radians, it doesn't matter much. The analysis for $\tan x=ax$ is quite similar to the one for $a=1$. – André Nicolas Jan 3 '13 at 20:45

I gave a talk on this equation in April 2006 and (a .pdf version of) the LaTeX slides I used may be of interest to you.

See also the math StackExchange question Derivation of asymptotic solution of $\tan (x)=x$. Finally, see the posts in this January 2006 sci.math thread: Regarding tan(x) = x.

There is no closed form for the solutions of $\tan(x) = x$, but let me state a few interesting facts. Let $(\lambda_{n})_{n \in \mathbb{N}}$ be the sequence that lists the positive solutions of $\tan(x) = x$ in increasing order. Then

• $\displaystyle \sum_{n=1}^{\infty} \frac{1}{\lambda_{n}} = \infty$.

• $\displaystyle \sum_{n=1}^{\infty} \frac{1}{\lambda_{n}^{2}} = \frac{1}{10}$.

• Guess Im not the only one who knows this :p +1 from me. – mick Jan 3 '13 at 20:56
• @mick: Oh, you mean the second identity? – Haskell Curry Jan 3 '13 at 20:59
• Yes ... What else ? – mick Jan 3 '13 at 21:00
• @HaskellCurry: Can u provide me some links to the explanations of this properties. Right now I am having difficulty understanding how these properties came. – Inquisitive Jan 3 '13 at 21:00
• You can refer to this other MSE thread: math.stackexchange.com/questions/75206/…. – Haskell Curry Jan 3 '13 at 21:10

$$\frac{\sin(x)}{\cos(x)}=x$$ $$\sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}...$$ $$\cos(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}...$$ Your question is equivalent to solving the equation

$$x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots=x-\frac{x^3}{2!}+\frac{x^5}{4!}-\cdots$$ $$x^3\left(\frac{1}{3!}-\frac{1}{2!}\right)-x^5\left(\frac{1}{5!}-\frac{1}{4!}\right)+x^7\left(\frac{1}{7!}-\frac{1}{6!}\right)+\cdots=0$$ Evidently giving $$x=0$$

The other solutions are given by the equation $$\left(\frac{1}{3!}-\frac{1}{2!}\right)-x^2\left(\frac{1}{5!}-\frac{1}{4!}\right)+x^4\left(\frac{1}{7!}-\frac{1}{6!}\right)-\cdots=0$$

But I don't know if there is a way to get from that to a closed form.