Sketching sets in the complex plane

Question

The question asks to sketch the subset of $\{z\ \epsilon\ C : -\frac\pi4 < arg(z) <\frac \pi 4 \}$

I know that $\frac \pi 4 \ $ is in the first quadrant but I'm unsure how to sketch this set

Answer 1

It is the set of complex numbers such that the polar angle is strictly between $-\frac{\pi}{4}$ and $\frac{\pi}{4}$.

So first, draw the two lines indicating those angles, that is the set of all complex numbers with these polar angles.

These lines are the half-lines $y=x$ and $y=-x$ from the origin to the right.

Then, the sector between them has such polar angles and only it does, so this is the required set (without the two lines since we have strict inequality)

Answer 2

You can best proceed by looking at the boundaries and an arbitrary point in the set; what do the three sets $$ S_{1}=\{ z\in\mathbb{C} : arg(z)=\frac{\pi}{4} \} $$

$$ S_{2}=\{ z\in\mathbb{C} : arg(z)=\frac{-\pi}{4} \} $$

$$ S_{3}=\{ z\in\mathbb{C} : arg(z)=0 \} $$ look like, and how do they relate to your set?