De Moivre's Theorem

Complex Numbers - Euler's Formula


Using Euler's formula \(e^{ix} = \cos x + i\sin x\), evaluate

\[\large e^{i \pi}.\]

Which of the following is equivalent to the conjugate of the complex number \(4e^{i\pi /4}?\)

The complex number \(z = -4 + 3i\) can be converted into the polar form \(z = re^{i\theta}.\)

What is the value of \(r?\)

If \(z = \sqrt{3} + i\), then what is the value of \(z^6?\)

Hint: You might want to start by converting \(\sqrt{3} + i\) into the form \(z = re^{i\theta}.\)

Given a complex number \(z = e^{ix}\), for which of the following values of \(x\) is the quantity \(\text{Re}(z)\text{Im(z)}\) minimized?


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