# Why does $\cos^{-1}\left(\cos(30)\right)$ (using radians) give the digits of $\pi$ after $3$?

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.

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 $$\cos^{-1}(\cos(30)) = \cos^{-1}(\cos(10\pi - 30)) = 10\pi - 30$$

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

See if you can prove it.

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 $$[-1,1]\to[0,\pi]$$.

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