Complex equation with arg(z), but unknown on the form a+bi

Question

$z=(x+i)^2$, $x > 0$ and a real number, solve for $x$

$\arg(z)=\frac {\pi}3$

$w = \sqrt z$

$w = x+i = \sqrt{x^2+1}$ $\ e^\frac{i\pi}{6}$

Now I could solve the bottom equation, but there must be an easier way? How should I have rather solved this question?

What do you want to solve for?? – Vishu Apr 28 '21 at 14:52 — Vishu, Apr 28 '21 at 14:52

score 1 · Accepted Answer · answered Apr 28 '21 at 15:38

1

You have $$x+i=\sqrt{x^2+1}e^{i{\arctan\frac{1}{x}}},$$ then $e^{i{\arctan\frac{1}{x}}}=e^{i\frac{\pi}{6}}$ which implies $$\arctan\frac{1}{x}=\frac{\pi}{6},$$ this gives you $$x=\cot\frac{\pi}{6}=\sqrt 3.$$

answered Apr 28 '21 at 15:38

janmarqz

10,538

Your $w$ is $w=\sqrt{3}+i $ then you can get $z=w^2$. – janmarqz Apr 28 '21 at 15:57
so $z=2+i2\sqrt{3}$. – janmarqz Apr 28 '21 at 22:50
thanks! man, glad to be helpful! – janmarqz Apr 29 '21 at 14:02

Complex equation with arg(z), but unknown on the form a+bi

1 Answers1