How can you find a complex number when you only know its argument?

If I know the limits to $\arg z$, how can I find the complex numbers those arguments stem from.

If I had to find $p$ and $$\frac{\pi}{4} < \arg ( z + p ) < \frac{3\pi}{4}$$ To solve this I need to find what complex numbers arg$\dfrac{\pi}{4}$ and arg $\dfrac{3\pi}{4}$ could stem from.

If I treat $\arg x$ (any argument) as the angle of a triangle made by the modulus and that $\arg$'s complex number then I can use the sin rule...

assume complex number is $a + bi$ then $$\dfrac{\pi}{4} =\sin ^ {-1}\frac{ b}{ \sqrt{b^2 + a^2} }$$

which simplifies to: $$0.5 ( b^2 + a^2 ) = b^2$$ which shows that $b^2 = a^2$, however I still don't know how to find $b$ (or $a$).

Is there a methord I am unaware of?

• Do you know about polar coordinates? If so, that's your easier method. Separately, the title suggests you are looking for a single number, while the question body tends to suggest you want to sketch a region defined by inequalities on the arg. As John Hughes says, knowing the arg of a complex number does not determine the number, only the ray on which it lies. By contrast, knowing an interval of angles easily allows you to sketch the set of complex numbers whose argument lies in the given interval. – Andrew D. Hwang Feb 3 '17 at 11:43

If you know that the argument of a complex number is $\pi/4$, for instance, it could be $1 + i$ or $2 + 2i$ or $23 + 23i$. So knowing the arg alone is not enough to determine the number, and no amount of cleverness will change that.
• No. For suppose that you knoe that $z = 0$. Then all of the answers I gave would still be valid ones (of the condition were less-that-or-equal; for the condition you've got, any point of the form $0 + s\mathbf i$ for any positive real $s$ would satisfy the equation). That's part of what I meant by "no amount of cleverness will change that." – John Hughes Feb 4 '17 at 12:31