De Moivre's Theorem

Complex Numbers - Euler's Formula


Using Euler's formula eix=cosx+isinxe^{ix} = \cos x + i\sin x, evaluate

eiπ.\large e^{i \pi}.

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

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

What is the value of r?r?

If z=3+iz = \sqrt{3} + i, then what is the value of z6?z^6?

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

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


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