State the principal argument interval used for a non-zero complex number \(z\).

Give one argument of the complex number \(3i\).

Find \(\operatorname{Arg}(-4)\).

Find \(\operatorname{Arg}(2-2i)\).

Find the principal argument of \(-3+3i\).

Find the principal argument of \(-5-5\sqrt{3}i\).

Write the general argument of \(1+i\) using an integer parameter.

Find the principal argument of \(\sqrt3-i\), giving the answer in radians.

A complex number has modulus \(6\) and principal argument \(-2\pi/3\). Find its Cartesian form.

Find all \(\theta\in[0,2\pi)\) such that \(\theta\) is an argument of \(-2i\).

Explain why \(\arg(z)\) is multi-valued but \(\operatorname{Arg}(z)\) is single-valued for \(z\ne0\).

A student says \(\operatorname{Arg}(-1-i)=5\pi/4\). Diagnose the error and give the correct principal argument.

Find \(\operatorname{Arg}\left((1+i)(-\sqrt3+i)\right)\) without first fully expanding the product.

Let \(z\) have \(\operatorname{Arg}(z)=3\pi/4\). Find the principal argument of \(-iz\).

For which real values of \(t\) is \(\operatorname{Arg}(t+i)=\pi/4\)?

For which real values of \(a\) does \(a-2i\) have principal argument \(-\pi/2\)?

Find all real \(x\) such that \(\operatorname{Arg}(x+xi)=3\pi/4\).

Prove that for any non-zero complex number \(z\), \(\operatorname{Arg}(-z)\) is obtained from \(\operatorname{Arg}(z)+\pi\) by adjusting into \((-\pi,\pi]\).

A non-zero complex number \(z\) satisfies \(\operatorname{Arg}(z)=\theta\). Determine \(\operatorname{Arg}(1/z)\) in terms of \(\theta\), including the endpoint case.

Show why \(\operatorname{Arg}(z_1z_2)\) is not always equal to \(\operatorname{Arg}(z_1)+\operatorname{Arg}(z_2)\), even though arguments add under multiplication.