6.4 De Moivre’s theorem

[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]?

Evaluate:

(a) [math]\left(2\left(\cos\dfrac{3\pi}{5}+i\sin\dfrac{3\pi}{5}\right)\right)^{-5}[/math]

(b) [math]\left(\dfrac{2}{1-i}\right)^{10}[/math]

Let [math]z[/math] be a complex number satisfying [math]|z|=3[/math] and [math]\arg(z)=\theta[/math]. Find the moduli and arguments of the following numbers.

(a) [math]\dfrac{1}{z^4}[/math]

(b) [math]iz^2[/math]

(c) [math]\dfrac{z^3}{i}[/math]

Find the square roots of the following numbers

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

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

Let [math]z_1, z_2, \cdots, z_8[/math] be the solutions of [math]x^8=1[/math] where [math]\arg(z_1)<\arg(z_2)<\arg(z_3)<\cdots<\arg(z_8)[/math]. If the area of polygon [math]z_1z_2z_3\cdots z_8[/math] is [math]\sqrt{a}[/math] where [math]a[/math] is an integer, what is the value of [math]a[/math]?

Solve the following equations. Express your answers in integers or fractions.

(a) [math]z^5=-32[/math]

(b) [math]z^{12}=-\dfrac{1}{2}+\dfrac{\sqrt{3}}{2}i[/math]

Find the minimum positive integer [math]n[/math] such that [math]3\left(\cos\frac{3\pi}{8}+i\sin\frac{3\pi}{8}\right)^n[/math] is a pure imaginary number.

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_1[/math] and [math]z_2[/math] be complex numbers. Suppose [math]|z_1|=2, |z_2|=3, \arg(z_1)=\dfrac{\pi}{12}[/math] and [math]\arg(z_2)=\dfrac{5\pi}{12}[/math]. Find the moduli and arguments of the following numbers. Express your answers in integers or fractions.

(a) [math]z^5_1[/math]

(b) [math]\dfrac{z^3_1}{z_2}[/math]

(c) [math]z_1z^*_2[/math]

(d) [math]z^{-4}_1z^3_2[/math]

Let [math]\omega[/math] be a complex solution of the equation [math]x^3=1[/math]

(a) Evaluate [math]\omega^2+\omega+1=0[/math].

(b) Evaluate [math](\omega+3)(\omega^2+3)(\omega^3+3)(\omega^4+3)(\omega^5+3)[/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]R[/math] be a pentagon with vertices [math]z_1, z_2, \cdots, z_5[/math]. These vertices are the solutions to the equation [math]x^5+x^4+x^3+x^2+x+1=0[/math]. If the area of [math]R[/math] is [math]a\sqrt{b}[/math] where [math]a[/math] is a rational number and [math]b[/math] is a prime number, what is the value of [math]ab[/math]?

Let [math]z[/math] be a complex number with modulus 1 and [math]\arg(z)=\dfrac{3\pi}{14}[/math]. Find the maximum negative integers [math]m[/math] and [math]n[/math] such that [math]z^m[/math] is a real number and [math]z^n[/math] is a pure imaginary number.

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