6.4 De Moivre’s theorem (in class)

[math](3-\sqrt{3}i)^6=-a^b[/math] where [math]a[/math] and [math]b[/math] are integers, what is the value of [math]b[/math]?

Find the square roots of the following numbers

(a) [math]-i[/math]

(b) [math]\dfrac{-1+\sqrt{3}i}{2}[/math]

Evaluate [math]\displaystyle{\sum^{17}_{k=1}\sin\left(\frac{2k\pi}{17}\right)}[/math].

Let [math]\omega[/math] be a nonreal solution of the equation [math]\omega^{24}=1[/math]. Evaluate [math]\displaystyle{\sum^{23}_{k=0}\omega^k}[/math]

Let [math]z[/math] be a complex number with modulus 1. Given that [math]\dfrac{1+z}{1+z^*}=z[/math], what is the value of [math]\left(\dfrac{1+\cos\frac{\pi}{3}-i\sin\frac{\pi}{3}}{1+\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}}\right)^3[/math]?
Note: [math]\dfrac{1+z}{1+z^*}=z[/math] is an identity. You can try to verify it.

Let [math]z[/math] be a complex number. Suppose [math]|z|=1[/math] and [math]\arg(z)=\theta[/math]
(a) Show that [math]z^n-z^{-n}=2i\sin\theta[/math].
(b) Show that [math]\sin3\theta=3\sin\theta-4\sin^3\theta[/math].