I came across something I cannot solve.
The exercise states:
Plot the values in the complex plane that satisfy the inequality:
$$\cos[{\arg{(-2iz^4)}}]\ge0$$
Here is what I did.
Step 1 - transform $-2iz^4$ into trigonometric and then exponetnial form so we can notice the argument easier. (I also raised the number to the power of 4)
$z^4 = \sqrt{x^2+y^2}^4(\cos{(4\arctan{\frac{y}{x}) +i\sin(4\arctan{\frac{y}{x})}}}$
Step 2 - multiply the number by $-2i$ which has the exponential form of $$2e^{i\LARGE{\frac{3\pi}{2}}}$$
by which we get that $-2iz^4$ is $ 2(x^2+y^2)^2\cdot e^({\frac{3\pi}{2}}+4\arctan{\frac{y}{x})}$
After which we can deduce that the argument of $cos$ in the original exercise is $$\frac{3\pi}{2} + 4\arctan{\frac{y}{x}}$$
Now I get $$\cos({\frac{3\pi}{2} + 4\arctan{\frac{y}{x}})} \ge0$$
This is the same as $$\sin({4\arctan{\frac{y}{x}})} \ge0$$
And I don't know how to proceed. I used Desmos grapher and I think that their plot is the same as the one I've got in my solutions in the workbook (leading me to think my work was correct until this point) but I have no clue how to proceed here.
What I tried to do: I tried applying the $\arcsin$ function to both sides and then applying $\arctan$ but what I get is $\frac{y}{x}\ge0$ which is not correct.
Can anyone advise me how to solve this? Thanks!