FAC1004 Tutorial 2 — Complex Numbers & De Moivre's Theorem

Centre for Foundation Studies in Science
Universiti Malaya
FAC1004 Advanced Mathematics II, 2025/2026

Question 1

Let $z_1 = -4 + 4i$ and $z_2 = -2\sqrt{3} + 2i$. Write the Cartesian form for $(z_1)^3$ and $(z_2)^6$.

Question 2

Simplify to terms in $\cos x$ and $\sin x$.

(a) $(\cos 7x + i \sin 7x)(\cos 5x - i \sin 5x)$

(b) $\frac{\cos 3x + i \sin 3x}{\cos 5x + i \sin 5x}$

Question 3

Express in both exponential form and Cartesian form.

(a) $2\left[\cos\left(\frac{2\pi}{3}\right) + i\sin\left(\frac{2\pi}{3}\right)\right]$

(b) $\frac{1}{\cos \frac{\pi}{4} + i\sin \frac{\pi}{4}}$

Question 4

Calculate:

(a) $\arg\left[\left(-\cos \frac{\pi}{4} - i\sin \frac{\pi}{4}\right)\left(-\cos \frac{\pi}{3} + i\sin \frac{\pi}{3}\right)\right]$

(b) The modulus and argument of $\frac{(1+i)^4}{(1-i)^2}$

Question 5

Show that $e^{ix} + e^{-ix} = 2\cos x$ and $e^{ix} - e^{-ix} = 2i\sin x$.

Deduce that for positive integer $n$:

(a) $e^{inx} + e^{-inx} = 2\cos(nx)$

(b) $e^{inx} - e^{-inx} = 2i\sin(nx)$

Hence, express $\cos(4x)$ as the sum of powers of $\cos x$ only and $\sin(4x)$ as the sum of product of powers of $\sin x$ and $\cos x$.

Question 6

Prove:

(a) $\cos(5\theta) = 16\cos^5 \theta - 20\cos^3 \theta + 5\cos\theta$

(b) $\sin(5\theta) = 16\sin^5 \theta - 20\sin^3 \theta + 5\sin\theta$

Question 7

Find the cube roots of the following numbers.

(a) $z = -3 + 3i$

(b) $z = 8\left(\cos \frac{2\pi}{3} - i\sin \frac{2\pi}{3}\right)$

Question 8

What are the tenth roots of unity? How many of these are on the real axis? Imaginary axis? In the first quadrants? Other quadrants?

Question 9

Find all $z$ for which $z^5 = -32i$ and $\text{Im}(z) > 0$.

Question 10

Suppose $\sqrt{3} - i$ is a root of $z^4 + 16(1+i)z^2 + a + ib = 0$, where $a$ and $b$ are real numbers. Find $a$ and $b$.

Question 11

Find the exact value of $\sqrt{z}$ for the following $z$. Determine the argument and modulus of each $\sqrt{z}$.

(a) $z = 1 + i$

(b) $z = -i$

Question 12

Find $z = x + iy$ for which:

(a) $z^2 = 2i$

(b) $z^3 = z + 2i$

Key Concepts Covered

Complex Numbers — Cartesian, polar, and exponential representations
De Moivre's Theorem — Powers and roots of complex numbers
Euler's Formula — Relationship between exponential and trigonometric functions
Roots of Unity — nth roots of complex numbers
Complex Conjugate — Properties and applications
Argument and Modulus — Magnitude and angle of complex numbers

Related Concepts

Complex Numbers