It seems like a problem of unclear or inapplicable definitions. As stated in the comments, it sounds like you've given the students the following definition: Given an angle $t$, we consider a right triangle with an angle of measure $t$, and define $\cos(t)$ to be the ratio of the side length of this triangle adjacent to the given angle to the length of the hypotenuse.
The problem is, this definition only makes sense when $0 < t < \pi/2$ (or, if you prefer degrees, $0^\circ < t < 90^\circ$. Indeed, a right triangle's other two angles have measure strictly between $0^\circ$ and $90^\circ$. But you've given the students a problem that assumes cosine has been defined for any $t$ between $0^\circ$ and $360^\circ$, which makes no sense with the definition they've been given!
I would recommend instead defining cosine as the $x$-coordinate on the unit circle. A simple diagram (drawing an appropriate right triangle inside the unit circle) shows this agrees with the other definition for angles in the first quadrant, but the unit circle definition also clearly makes sense for any angle. Plus, it makes it much easier to reason about this kind of question — just draw the line $x = 1/2$ and see where it intersects the unit circle, then use geometric reasoning to find the angles.
I'm still a bit confused about what you mean by "algebraic methods", by the way. Both the triangle definition and the unit circle definition are geometric, so you really can't compute values of cosine using these definitions without appealing to geometry.