How to solve $\cos x = x \sin x$

Question

Is it possible to express the solutions to $\cos x = x \sin x$ in closed form?

Numerically, the first positive solution seems to be $x = 0.8603335890193...$, which is is suspiciously close to $\frac{\sqrt{3}}{2} = 0.8660254037844...$.

Answer 1

The solutions are the fixed points of $x \mapsto \cot x$. I'm not aware of any closed-form expressions for them.

Answer 2

Suspicion could start after agreement with about 6th decimal place or so. Numerical solution should be resorted to because analytical solution is not possible. Approximate solution is possible at their intersection by graphing the two functions .

Answer 3

A closed form does not exist (remember that this is already the case for $x=\cos(x)$).

Since you are obviously considering the first root of the equation, we can build good approximations.

Consider around $x=1$ $$f(x)=\cos( x) -x \sin (x)=$$ $$\cos (1)-\sin (1)+\sum_{n=1}^\infty \frac{(n+1) \cos \left(\frac{\pi n}{2}+1\right)-\sin \left(\frac{\pi n}{2}+1\right)}{n!}(x-1)^{n}$$

Truncate to any order and use series reversion and you will find $$x=1+t+\frac{ \sin (1)-3 \cos (1)}{2 (2 \sin (1)+\cos (1))}t^2+\frac{ 39-12 \sin (2)+17 \cos (2)}{12 (2 \sin (1)+\cos (1))^2}t^3+O\left(t^4\right)$$ where $$t=-\frac{f(x)+\sin (1)-\cos (1)}{2 \sin (1)+\cos (1)}$$ Making, as desired, $f(x)=0$ and, using multiple angle formulae, we end with the approximation $$x=\frac {934 \sin (1)-27 \sin (3)-185 \sin (5)+1036 \cos (1)-744 \cos (3)-44 \cos (5) } {48 (2 \sin (1)+\cos (1))^5}$$ which, numerically, is $0.860439$.

For sure, this can be improved using more terms; for example, expanding up to $O((x-1)^6)$, we should get $x=0.8603343$ while the "exact" value you reported is $x=0.8603336$.

Now, even if it does not mean anything, if you want a nice looking number, the inverse symbolic calculator proposes $$x \sim \sqrt{ \frac{3 \sqrt[3]{3}}{8-\sqrt[3]{10}} }=0.8603350$$ For this value $f(x)=-2.88\times 10^{-6}$.