How do you calculate arccos(cos(4))?

Apr 26, 2015

$\arccos \left(\cos \left(4\right)\right)$ is a value, $t$ between $0$ and $\pi$ whose cosine is equal to the cosine of $4$.

Note that $4 > \pi$, but $4 \le \frac{3 \pi}{2} \approx \frac{9.42}{2} = 4.71$.

Thinking of $t = 4$ as the radian measure of an angle,
it corresponds to an angle in quadrant 3. and the reference angle is $4 - \pi$.

The second quadrant angle with the same reference angle is $\pi - \left(4 - \pi\right)$ , which is $2 \pi - 4$

So
$\arccos \left(\cos \left(4\right)\right) = 2 \pi - 4$

Note
All of the above is assuming that you did not intend to ask for $\arccos \left(\cos \left({4}^{\circ}\right)\right)$ , which, of course, is ${4}^{\circ}$.