Cyclometric equation $x\arcsin x=1$

Question

What method should I use to solve following equation for $x$ $$ x\arcsin x=1? $$ I would be grateful for any comment.

Answer 1

$$x\arcsin x = 1 \implies \arcsin x = \frac{1}{x} \implies \sin \frac{1}{x} = x$$

There isn't a good way to solve this, so I guess you can estimate using the MacLaurin expansion of $\sin x$.

$$\sin x = x-\frac{x^3}{3!}+\frac{x^5}{5!}-... \implies \sin \frac{1}{x} = \frac{1}{x}-\frac{1}{3!x^3}+\frac{1}{5!x^5}-...$$

You can truncate the series at a point to estimate for $x$. For example, you can solve

$$\frac{1}{x}-\frac{1}{3!x^3} = x$$

which gives $x \approx \pm0.888$ (true) and $x \approx \pm0.46$ (extraneous). You can use more terms for a more accurate estimate.