In order for arctan to be a function, arctan(-1) must have just one value, and the same has to be true for arctan(x), no matter what real number x stands for.
If arctan's range consisted of angles in QI, QII, and QIV, then what's arctan(-1)? There's an angle in QII, namely 135 degrees, whose tangent is -1, and there's an angle in QIV, namely 315 degrees (or -45 degrees, if you prefer) whose tangent is -1.
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