How to solve $x = \arccos(x)$

Question

How does one solve $$ x = \arccos(x) $$ for $x$?

Is there an exact solution achievable by hand? I tried wolfram|alpha, but it only spits out the solution $$ x = \text{ root of }\quad x-\cos^{-1}(x) \quad \text{ near } x = 0.739085 $$

Think about the relationship between solving $x = \operatorname{arccos}(x)$ and $\cos x = x$. — hardmath, Jul 02 '14 at 01:14

score 1 · Accepted Answer · answered Jul 02 '14 at 00:51

1

There is no solution to this equation in terms of elementary functions. However, one can still use numerical methods to solve it (which is probably how Wolfram|Alpha generates the solution).

answered Jul 02 '14 at 00:51

zeta

273

Thanks! I guess this renders my work kind of useless... – decoromath Jul 02 '14 at 01:11

How to solve $x = \arccos(x)$

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