Recently I encounter a problem (of trigonometry) where $\sin{x}$ was asked and it was also told that $x$ is acute.
So, I like anyone else, found the general solution, however my general solution provide me with the values of $x$ in all the $4$ quadrants and obviously it also had values exceeding $2\pi$..
[in my lower class I was taught that angles $<90^{\circ}$ are acute and $>90^{\circ}$ are obtuse, but they never went beyond $180^{\circ}$]
My doubt: while writing the answer, would we
$1.$ consider angles $<90^{\circ}$ degrees as acute, or
$2.$ angles in first quadrant as acute?
And what for Obtuse?
Also, my question is not only restricted to trigonometric functions.
AND when the angle is greater than 360 degrees should we take mod360 or not?
Acute angles are angles between $0^\circ$ and $90^\circ$.
Obtuse angles are angles between $90^\circ$ and $180^\circ$.
Reflex angles are angles between $180^\circ$ and $360^\circ$.
As far as I am aware, angles of size greater than $360^\circ$ do not have a special name.
I believe these names exist because of reference to the angles inside polygons, which is why negative angles and angles beyond $360^\circ$ do not have names.
The terminology "acute angle" refers specifically to these sizes and is not equivalent to saying "in the first quadrant". Angles in the first quadrant are any angles between $0^\circ$ and $90^\circ$, or between $360^\circ$ and $450^\circ$, or between $720^\circ$ and $810^\circ$, etc.
So in the context of the problem you have, it is most likely specifically asking for angles between $0^\circ$ and $90^\circ$ and not just in the first quadrant.
To answer the title of the question, $361^\circ$ is neither acute nor obtuse.