View on the complex plane a set of points that satisfy the conditions $|\pi -\arg(z)|>\frac \pi 4$

Question

My task: $$|\pi -\arg(z)|>\frac \pi 4$$

I think should be used the formula $z=x+iy$, or not? In general, the need to bring in a normal type and draw the graph. Thank you.

Answer 1

Writing $z = x+iy$ is unnecessary for solving this by hand. We can rearrange your inequality as $\arg(z)<\frac{3\pi}{4}$ or $\arg(z)>\frac{5\pi}{4}$, which is true for all points outside of the WNW (west-north-west) octant of the plane (don't forget that $\arg$ is usually, but not always, defined from $\mathbb{C}$ to $(-\pi, \pi]$).