Can the argument of a complex number be negative? If yes, what is meant by a negative argument if observed graphically?

Question

Please find the modulus and argument of the complex number given by

$$z = \frac{(1 + cosθ + isin)^5}{cos3θ + sin3θ}$$

My solution :-

$$z = (2cos^2\frac{θ}{2} + isinθ)^5(cos3θ - isin3θ)$$ $$z = ((2cos\frac{θ}{2})(cos\frac{θ}{2} + isin\frac{θ}{2}))^5(cos3θ - isin3θ)$$ $$z = 32cos^5\frac{θ}{2}(cos(\frac{5θ}{2} - 3θ) + isin(\frac{5θ}{2} - 3θ))$$ $$z = 32cos^5\frac{θ}{2}(cos(-\frac{θ}{2}) + isin(-\frac{θ}{2})$$

$$Argument = -\frac{θ}{2}$$ $$Modulus = 32|cos^5\frac{θ}{2}|$$

So the argument can be negative? Is my solution wrong? What is meant by argument being negative? Isn't argument supposed to a measure of counter clock wise rotation from the positive x - axis?

Thank you

Answer 1

It is completely fine (and preferred in some cases) to have a negative argument. A negative counterclockwise rotation becomes a clockwise rotation, that's all.