Polar, exponential form of Complex numbers (conversions, arithmetics), combining phase shifts, drawing on argand diagram: notes and practice questions

In a complex signal processing system, an input signal $z$ is transformed into an output signal $w$ using the relationship $w = -iz + (3 + 2i)$, where $w, z \in \mathbb{C}$.

(a.i) Calculate $w$ when $z = 4 + i$.

(a.ii) Calculate $w$ when $z = 5$.

(b) Describe the two geometric transformations that map point $z$ to point $w$ on the Argand diagram, stating the order in which they are applied.

(c) Find the complex number $z$ such that the output signal $w = 1 - i$.

(a) A complex number $w$ represents a displacement on the complex plane, given by $w = -3 + 3i$. Determine the magnitude and direction (argument) of this displacement.

(b) Another complex number $z$ represents a scaling and rotation, given in polar form as $z = 2\left(\cos\left(\frac{2\pi}{3}\right) + i \sin\left(\frac{2\pi}{3}\right)\right)$. A combined transformation is defined by $T = w^3 z^2$. Find $T$ in its simplest polar form.

Let $w_1$ be a complex number representing a rotation and scaling transformation, given by $w_1 = 2 \text{ cis} \left(\frac{2\pi}{3}\right)$.

(a) In parts (a)(i) and (a)(ii), give your answers in the form $re^{i\theta}, r\geq 0, -\pi < \theta\leq \pi$.

(i) Calculate $w_1^4$.

Another transformation $w_2$ is given by $w_2 = 4 \text{ cis}\left(\frac{k\pi}{9}\right)$, where $k \in \mathbb{Z}^+$.

(ii) For $k=3$, calculate $\left(\frac{w_1}{w_2}\right)^3$.

(b) Find the least positive integer value of $k$ such that the combined transformation $w_1 w_2$ results in a purely positive scaling (i.e., $w_1 w_2 \in \mathbb{R}^+$).

A specialized navigation chart uses complex numbers to represent locations. The central hub is at the origin $(0,0)$. Two important landmarks, 'Anchor Point' ($z_A$) and 'Beacon Tower' ($z_B$), are located at $z_A = 5 - 2i$ and $z_B = 4e^{i\frac{\pi}{3}}$ respectively.

(a) (i) Find the modulus of $z_A$.

(a) (ii) Find the argument of $z_A$, giving your answer in radians.

(b) Find the area of the triangular region formed by the central hub, Anchor Point ($z_A$), and Beacon Tower ($z_B$).

(a) The trajectory of a probe in a simulated complex plane is given by the complex number $z_\theta = \frac{1}{3}\theta e^{i\frac{\theta}{2}}$, where $\theta \ge 0$ is a parameter representing time. An Argand diagram is provided showing a circular boundary with radius 5 units.

(a.i) Plot on the Argand diagram the point corresponding to $\theta = \pi$.

(a.ii) Plot on the Argand diagram the point corresponding to $\theta = 2\pi$.

(a.iii) Plot on the Argand diagram the point corresponding to $\theta = 3\pi$.

(b) The probe exits a stable orbital region when its distance from the origin, $|z_\theta|$, reaches 5 units.

(b.i) Calculate the value of $\theta$ when the probe exits the stable orbital region.

(b.ii) For this value of $\theta$, plot the approximate position of $z_\theta$ on the Argand diagram.

In a system analyzing oscillating signals, the initial state of a signal is given by the complex number $w = 1 - i\sqrt{3}$.

(a) Express $w$ in the form $re^{i\theta}$, where $-\pi < \theta \le \pi$.

After a certain operation, the signal's state changes from $w$ to $w^2$.

(b) (i) Describe fully the single transformation (a composition of an enlargement and a rotation about the origin) that maps the point representing $w$ to the point representing $w^2$ on the Argand diagram. State the scale factor of the enlargement and the angle of rotation.

(b) (ii) This transformation can be represented by a matrix. Find and simplify the $2 \times 2$ matrix that represents this transformation.

The signal is considered stable when its state $w^n$ is real and positive.

(c) Find the smallest positive integer, $n$, for which $w^n$ is real and positive.

Two sound waves propagate through a medium. The pressure variation, $P_1$, from the first wave at a specific point, at time $t$, is modelled by

$P_1 = Re (3e^{2ti})$.

The pressure variation, $P_2$, from the second wave at the same point is modelled by

$P_2 = Re (4e^{(2t+0.5)i})$.

The total pressure variation at the point, $P_T$, is given by

$P_T = P_1 + P_2$.

Find an expression for $P_T$ in the form $A \cos(Bt + C)$, where $A, B$ and $C$ are real constants.

Hence write down the maximum pressure variation in the medium.

A complex number is defined by $z = \sin(2\theta) - \text{i}(1 + \cos(2\theta))$ for $-\frac{\pi}{2} \le \theta \le \frac{\pi}{2}$.

(a) Solve $3 \cos(x - 45^{\circ}) = \sin(x + 45^{\circ})$ for $0^{\circ} \le x \le 360^{\circ}$.

(b) Show that $\cos 15^{\circ} - \sin 15^{\circ} = \frac{1}{\sqrt{2}}$.

(c) Find the modulus and argument of $z$ in terms of $\theta$. Express each answer in its simplest form.

(d) Hence find the fourth roots of $z$ in modulus-argument form.

The depth of water at high tide, $H$ meters, in Port Azure can be modelled by

$H = 1.5 \cos (0.0165t + 0.5) + 8.0$,

where $t$ is the day of the year 2024 (for example, $t = 1$ represents 1 January 2024).

The depth of water at low tide, $L$ meters, in Port Azure can be modelled by

$L = 1.0 \cos (0.0165t - 2.5) + 6.0$.

The tidal range, $R$ (the difference between high and low tide depths), in Port Azure during 2024 can be modelled by

$R = a \cos (0.0165t + b) + c$.

Find the value of $a$, of $b$ and of $c$.

Hence, or otherwise, find the largest tidal range in Port Azure during 2024

and the day of the year on which this occurs.

Two loudspeakers are emitting sound waves of equal frequencies in a room. The pressure variation from the first loudspeaker is modelled by the equation $P = 40 \sin(t + 75\degree)$, where $P$ is in Pascals (Pa) and $t$ is in seconds.

The pressure variation from the second loudspeaker is modelled by the equation $P = 50 \sin (t + 20\degree)$.

Determine the maximum pressure variation of the combined sound waves.

Using your graphic display calculator, find a suitable equation for the combined pressure variations, giving your answer in the form $P = P_0 \sin(at + b)$, where $a$, $b$ and $P_0$ are constants, $a > 0$ and $0\degree \le b < 180\degree$.

In an AC circuit, the impedance of a component is given by the complex number $Z = 1 - i\sqrt{3}$ ohms.

Write $Z$ in the exponential form $re^{i\theta}$, where $r \in \mathbb{R}^+$ and $-\pi < \theta \leq \pi$.

Two current phasors in a parallel circuit are given by $I_1 = e^{3ti}$ and $I_2 = 3e^{\left(3t+\frac{\pi}{4}\right)i}$. The total instantaneous reactive current is given by the imaginary part of their sum, $\text{Im}(I_1 + I_2)$.

Find $\text{Im}(I_1 + I_2)$ in the form $p \sin(3t + q)$, where $p > 0$, $t \in \mathbb{R}$ and $-\pi \leq q \leq \pi$.

In an alternating current (AC) circuit analysis, complex numbers are often used to represent impedances. Consider two impedances, $Z_A$ and $Z_B$, given in polar form as:

$Z_A = 6 \text{ cis}\left(\frac{2\pi}{3}\right)$

$Z_B = 2 \text{ cis}\left(-\frac{\pi}{6}\right)$

Find an expression for $Z_A Z_B$. Give your answer in the form $a + bi$, where $a,b \in \mathbb{R}$.

Find an expression for $\frac{Z_A}{Z_B}$. Give your answer in the form $a + bi$, where $a,b \in \mathbb{R}$.

Find an expression for $(Z_B)^3$. Give your answer in the form $a + bi$, where $a,b \in \mathbb{R}$.

A company models the position of two drones, 'Alpha' and 'Bravo', on a complex plane relative to a central control tower located at the origin. The position of Drone Alpha is given by $z_A = 1 - i$ and the position of Drone Bravo is given by $z_B = -\sqrt{3} + i$.

(a) Express $z_A$ and $z_B$ in modulus-argument (polar) form.

(b) Determine the complex number representing the combined effect of their positions, given by $z_A z_B$, in polar form.

(c) Determine the complex number representing the relative position of Drone Alpha with respect to Drone Bravo, given by $\frac{z_A}{z_B}$, in polar form.

(d) Predict the position of Drone Alpha after a specific command that scales its argument by a factor of 4 and raises its modulus to the power of 4. This new position is given by $(z_A)^4$. Express this in polar form and then in Cartesian form.

Consider two alternating current (AC) voltage sources connected in series. The instantaneous voltages, $V_1(t)$ and $V_2(t)$, are given by the functions:

$V_1(t) = 20.0 \sin(0.40t + 0.85)$

$V_2(t) = 15.0 \sin(0.40t - 1.50)$

where $t$ is time in seconds and voltages are in volts.

(a) Write down both functions in the form $\text{Im}(Ae^{i\omega t} e^{i\phi})$.

(b) The total instantaneous voltage across the series combination is $V_{total}(t) = V_1(t) + V_2(t)$, which can be expressed in the form $V_{total}(t) = q \sin(0.40t + r)$, where $0 \le r < 2\pi$.

Use your answer to part (a) to find the value of $q$ and of $r$. Give your answers to three significant figures.

Let $u = be^{\frac{\pi}{3}i}$, where $b\in\mathbb{R}^+$.

For $b = 3$,

(i) find the values of $u^2, u^3$, and $u^4$;

(ii) draw $u, u^2, u^3$ and $u^4$ on the following Argand diagram.

.

Let $v = \frac{u}{1+2i}$.

Find the value of $b$ for which successive powers of $v$ lie on a circle.

In an alternating current (AC) circuit, two impedances are given by the complex numbers $z_{1} = 3 - 3\sqrt{3}\text{i}$ and $z_{2} = -4 - 4\text{i}$. The total impedance is given by $w = \frac{z_{1}}{z_{2}}$.

(a) (i) By expressing $z_{1}$ and $z_{2}$ in modulus-argument form, write down the modulus of $w$.

(a) (ii) Write down the argument of $w$.

(b) The circuit is said to be purely resistive when $w^n$ is a real number. Find the smallest positive integer value of $n$ for which this occurs.