6.4 De Moivre’s theorem practice 1

Evaluate:

(a) $\left(2\left(\cos\dfrac{3\pi}{5}+i\sin\dfrac{3\pi}{5}\right)\right)^{-5}$

(b) $\left(\dfrac{2}{1-i}\right)^{10}$

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

(a) $\dfrac{1}{z^4}$

(b) $iz^2$

(c) $\dfrac{z^3}{i}$

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

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

(a) $z^5=-32$

(b) $z^{12}=-\dfrac{1}{2}+\dfrac{\sqrt{3}}{2}i$

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

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

(a) $z^5_1$

(b) $\dfrac{z^3_1}{z_2}$

(c) $z_1z^*_2$

(d) $z^{-4}_1z^3_2$

Let $\omega$ be a complex solution of the equation $x^3=1$

(a) Evaluate $\omega^2+\omega+1=0$.

(b) Evaluate $(\omega+3)(\omega^2+3)(\omega^3+3)(\omega^4+3)(\omega^5+3)$.

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

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