Question #40416

This question tests the intersection of conditional trigonometric identities and the domain restrictions of inverse trigonometric functions.

Students often blindly apply the heuristic $\arcsin(\sin x) = x$ and $\arccos(\cos x) = x$, treating them as purely algebraic cancellations.

By forcing the angles into specific quadrants via the identity $\cos(3A)+\cos(3B)+\cos(3C)=1$ (which requires one angle to be $2\pi/3$), the triple angles $3A$ and $3B$ fall outside the principal domains $[-\pi/2, \pi/2]$ and $[0, \pi]$.

This creates a 'continuity rupture' where the composite functions must be evaluated using their periodic definitions, trapping students who ignore the piecewise nature of inverse compositions.