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I was messing around with my calculator when in radian mode I entered the following

$$\cos^{-1}\left(\cos(30)\right)$$

and it gave back $1.4159$. Basically, the digits of pi after $3$.

Is this merely a coincidence or is there something more to this? It seems kinda interesting.

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If $0\leq x\leq \pi$, then we have $\cos^{-1}(\cos(x)) = x$. That's basically the definition of the inverse cosine.

Now, adding and subtracting integer multiples of $2\pi$ to the argument of the cosine doesn't change the value, and changing the sign of the argument doesn't change the value of the cosine either. So we have $$\cos(30) = \cos(30 - 10\pi) = \cos(10\pi - 30)$$ Finally, noting that $0\leq 10\pi - 30\leq \pi$, we see that this means $$ \cos^{-1}(\cos(30)) = \cos^{-1}(\cos(10\pi - 30)) = 10\pi - 30 $$

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$\arccos(\cos(30)) = -30 + 10 \pi$.

See if you can prove it.

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All of the possible values of $\cos^{-1}\cos \theta$ are $\pm\theta + 2\pi k$ for integral $k$.

Your calculator chooses the value that is in $[0,\pi]$ so that its function $\cos^{-1}$ is a well-defined continuous single-valued mapping $[-1,1]\to[0,\pi]$.

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  • $\begingroup$ integral = integer $\endgroup$ – Nick Jan 9 '20 at 14:47
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    $\begingroup$ @Nick : Yes, that’s what “integral” means here (as opposed to, say, rational, transcendental, real, etc). It is an adjective. $\endgroup$ – MPW Jan 9 '20 at 15:04

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