Question #40417

This question tests the student's ability to map algebraic degree inputs to the principal value branches of inverse trigonometric functions.

The primary trap lies in assuming a false periodicity (e.g., 180 instead of 360) or failing to correctly identify the piecewise definitions of $\sin^{-1}(\sin \theta)$ and $\cos^{-1}(\cos \theta)$ across different quadrants.

A common mistake is assuming $\sin^{-1}(\sin x^\circ) = \frac{\pi x}{180}$ universally, which yields no solutions, or finding only the first branch solution ($x=105$) and missing the fourth quadrant branch ($x=345$).

Another trap is assuming the combination has a period of 180, leading to the incorrect sum of 2325.