# How do you evaluate arccos(sqrt3/2) without a calculator?

$\arccos \left(\frac{\sqrt{3}}{2}\right) = {30}^{\circ} \left(= \frac{\pi}{6} \text{ radians}\right)$
If we split an equilateral triangle (in which all interior angles $= {60}^{\circ} \mathmr{and} \frac{\pi}{3} \text{ radians}$) into two triangles as indicated:
we will get an angle $= \frac{{60}^{\circ}}{2} = {30}^{\circ}$ whose $\cos$ is $\frac{\sqrt{3}}{2}$
(Note that there is also the angle ${330}^{\circ}$ which gives this $\frac{\sqrt{3}}{2}$ ratio, but by the standard definition of ${\cos}^{- 1}$ as a function we are restricted to the range $\left[0 , \pi\right]$