I am given the complex number:
$$z = \bigg ( \dfrac{\sqrt{3} – i}{1 + i} \bigg ) ^ {12}$$
And I have to choose a true description of this number. Only one of the following descriptions is true:
A. $z = 2^6$
B. $\arg(z) = \pi$
C. $|z| = 2 ^ {12}$
D. $z = 64i$
E $\arg(z) = 2 \pi$
My problem is that I can't manipulate the number $z$ such that I can use DeMoivre's formula on $z$. This is as far as I got:
$$z =
\bigg ( \dfrac{\sqrt{3} – i} {1 + i} \bigg ) ^ {12} =
\bigg ( \dfrac{(\sqrt{3} – i) (1 – i)} {1 – i^2} \bigg ) ^ {12} =
\bigg ( \dfrac{\sqrt{3} – \sqrt{3}i – i + i^2} {2} \bigg ) ^ {12} =$$
$$ = \bigg ( \dfrac{\sqrt{3} – 1 – (\sqrt{3} + 1)i} {2} \bigg ) ^ {12}
= \bigg ( \dfrac{\sqrt{3} – 1} {2} + \dfrac{-(\sqrt{3} + 1)}{2} \bigg ) ^ {12}$$
And this is where I got stuck. I know that I need to get $z$ in a form that looks something like this:
$$z = (\cos \theta + i \sin \theta) ^ {12}$$
But I can't find an angle whose cosine equals $\dfrac{\sqrt{3} – 1}{2}$ and whose sine equals $\dfrac{-(\sqrt{3} + 1)}{2}$. So how can find the following:
$$\cos \hspace{.1cm} ? = \dfrac{\sqrt{3} – 1}{2}$$
$$\sin \hspace{.1cm} ? = \dfrac{-(\sqrt{3} + 1)}{2}$$
Best Answer
The key insight is to recognize roots of unity in the expression.
Let $a=1+i$, $b=\sqrt{3}-i$. Let $A=a/\sqrt 2$, $B=b/2$.
Then $A^4=-1$, $B^6=-1$, and so $$ z=\frac{b^{12}}{a^{12}} =\frac{2^{12} B^{12}}{2^6 A^{12}} =-2^{6} $$