# Math Help - Complex number with De Moivre's formula

1. ## Complex number with De Moivre's formula

don't know where should i post this.

Find the real part of the complex number (i-1)^3 by using De Moivre's formula

My lecture notes is too brief need help !

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2. Write it in the polar form:

$\displaystyle (i-1)^3 = \bigg[\sqrt{2}\left(\cos\left(\frac{3\pi}{4}\right)+i\si n\left(\frac{3\pi}{4}\right)\right)\bigg]^3$.

De Moivre's theorem tells you that, for any arbitrary $x$ and integer $n$, we have:

$\left(\cos{x}+i\sin{x}\right)^n = \cos\left(nx\right)+i\sin\left(nx\right)$.

3. Expand $\left( { - 1 + i} \right)^3 = \sum\limits_{k = 0}^3 {\dbinom{3}{k}\left( { - 1} \right)^{3 - k} \left( i \right)^k }.$
The real parts occur when $k=0~\&~2$.
What do they add up to?