Q.Find the modulus and the principal argument of $z=-1+i$, and express it in polar form.

The point lies in Quadrant II, so use $\theta=\pi-\alpha$ with reference angle $\alpha=\tan^{-1}|b/a|$.

Here $a=-1,\ b=1$.

Modulus:

Argument: the point $(-1,1)$ has $a<0,\ b>0$, so it lies in Quadrant II. The reference angle is

and in Quadrant II the principal argument is

Polar form:

$z=\sqrt2\left(\cos\frac{3\pi}{4}+i\sin\frac{3\pi}{4}\right).$ …

Quoting $\theta=\tan^{-1}(1/-1)=-\pi/4$ straight from the calculator — that points into Quadrant IV, the wrong direction. For …