How to find the main argument of a complex number?

Question

Mathematics

Hello!
To find the argument of a complex number, I use the formulas: cos ϕ = x / √ (x^2 + y^2), sin ϕ = y / √ (x^2 + y^2), for the real and imaginary parts of the number, then I get ϕ values.
Example:
z = -1
cos ϕ = 0, ϕ= π/2 + πn
sin ϕ = -1, -π/2 + 2πn
However, the textbook answers: arg(-1 )= π There is
only one main argument, please tell me how to find and select it?

Answer 1

All values ​​of the argument that differ by 2pi*n give the same complex number. All answers are equal. Your way of writing the answer is formally more correct, but apparently the author of the textbook considered this obvious, and chose one point.