State the polar form of a non-zero complex number \(z=a+bi\).

Find the modulus of \(6+8i\).

Convert \(2\operatorname{cis}(\pi/3)\) to Cartesian form, where \(\operatorname{cis}\theta=\cos\theta+i\sin\theta\).

Write \(-5\) in polar form using the principal argument.

Write \(3+3i\) in polar form using the principal argument.

Write \(-2+2\sqrt3 i\) in polar form using the principal argument.

Convert \(4\operatorname{cis}(-\pi/6)\) to Cartesian form.

Write \(-i\sqrt7\) in polar form using a positive modulus and principal argument.

Convert \(-6-6i\) to polar form using the principal argument.

Convert \(5\operatorname{cis}(7\pi/6)\) to Cartesian form and then rewrite it with a principal argument.

Explain why \(r\) is required to be non-negative in the usual polar form \(z=r\operatorname{cis}\theta\).

A student writes \(-1+\sqrt3 i=2\operatorname{cis}(-\pi/3)\). Find and correct the error.

Find the polar form of \(\dfrac{1+i}{\sqrt3-i}\) using moduli and arguments.

A complex number has polar form \(8\operatorname{cis}\theta\) and Cartesian form \(a+bi\). If \(\theta=2\pi/3\), find \(a\) and \(b\).

For which real \(t\) does \(t+2i\) have modulus \(4\)?

For which real \(t\) does \(t+i\sqrt3\) have principal argument \(\pi/3\)?

Find all real \(t\) such that \(t-ti\) has modulus \(3\sqrt2\) and principal argument \(-\pi/4\).

Prove from Cartesian coordinates that \(|z|=r\) when \(z=r(\cos\theta+i\sin\theta)\) and \(r\ge0\).

Show that \(r\operatorname{cis}\theta=r\operatorname{cis}\phi\) with \(r>0\) implies \(\theta-\phi=2k\pi\) for some integer \(k\).

A student converts \(-\sqrt3-i\) to polar form as \(2\operatorname{cis}(\pi/6)\). Give a complete diagnosis and correction.