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.
$\endgroup$3 Answers
$\begingroup$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$$ \cos^{-1}(\cos(30)) = \cos^{-1}(\cos(10\pi - 30)) = 10\pi - 30 $$
$\endgroup$ $\begingroup$$\arccos(\cos(30)) = -30 + 10 \pi$.
See if you can prove it.
$\endgroup$ $\begingroup$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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