Exercises: Operations with complex numbers in polar form

Given the complex numbers $z_1 = 4_{30^\circ}$ and $z_2 = 3_{60^\circ}$, calculate their product $z_1 \cdot z_2$, expressing the result in polar form.

Calculate the quotient $\frac{12_{150^\circ}}{3_{45^\circ}}$ and express the result in polar form.

Using de Moivre's formula, calculate the power $(2_{40^\circ})^3$ and express the result in polar form.

Find the multiplicative inverse of the complex number $z = 5_{120^\circ}$, expressing the final result in polar form.

A ship sails in the direction and distance indicated by the vector associated with the complex number $z = 10_{30^\circ}$. Due to weather conditions, the captain decides to turn $90^\circ$ anticlockwise while maintaining the same distance. By which complex number in polar form must $z$ be multiplied to obtain its new trajectory, and what is the final position?

Calculate the two square roots of the complex number $16_{120^\circ}$ and express the results in polar form.

Perform the following combined operation and express the result in polar form:

Given the complex number $z = 3_{50^\circ}$, express its conjugate $\bar{z}$ and its opposite $-z$ in polar form.

Carmen is solving an exercise on powers of complex numbers in polar form and writes the following steps in her notebook: $(2_{30^\circ})^5 = (2 \cdot 5)_{30^\circ \cdot 5} = 10_{150^\circ}$.

1. a) Identify and explain, with reasoning, the conceptual error made by Carmen when calculating the modulus. Justify geometrically why, in De Moivre's formula, the modulus is raised to the power and not multiplied by the exponent.
2. b) Correct the calculation by writing the correct result in polar form.
3. c) Convert the correct result into binomial form and indicate in which quadrant of the complex plane its affix is located.

Given the complex numbers $z_1 = 4_{45^\circ}$ and $z_2 = 2_{120^\circ}$:

1. a) Before calculating, make a reasoned estimate: in which quadrant of the complex plane will the product $z_1 \cdot z_2$ lie? Will its distance to the origin be greater or smaller than that of $z_1$? Reason similarly for the quotient $\dfrac{z_1}{z_2}$.
2. b) Check your estimates by calculating the exact product $z_1 \cdot z_2$ and quotient $\dfrac{z_1}{z_2}$ in polar form, reducing the arguments to the interval $[0^\circ, 360^\circ)$.
3. c) From the result of the quotient, calculate its inverse $\left(\dfrac{z_1}{z_2}\right)^{-1}$ in polar form and check whether its modulus is $0,5$ and whether its argument coincides with the one you would obtain by calculating $\dfrac{z_2}{z_1}$ directly.

Solve the equation $z^4 + 81 = 0$ in the set of complex numbers. Express the solutions in polar form and represent them graphically in the complex plane (real axis horizontal and imaginary axis vertical).

In an alternating current circuit, the total impedance $Z_T$ of two components connected in parallel is calculated using the formula $\frac{1}{Z_T} = \frac{1}{Z_1} + \frac{1}{Z_2}$. If the impedances of the components are $Z_1 = 10_{30^\circ} \ \Omega$ and $Z_2 = 10_{-30^\circ} \ \Omega$, find the total impedance $Z_T$ in polar form. Hint: convert to binomial form to perform the addition.

Solve the equation $z^3 = -8i$. First convert the number $-8i$ to polar form before calculating the roots, and express the final results in both polar form and binomial form.

Calculate the value of $(1 - \sqrt{3}i)^5$. To do this, first transform the base into polar form, apply de Moivre's formula and give the final result in binomial form.

Determine the smallest positive integer $n$ such that the result of $(\sqrt{3} + i)^n$ is a negative real number.

An underwater exploration drone starts from the origin and moves in a straight line to the point corresponding to the affix of the complex number $z_1 = 2_{45^\circ}$ (in kilometres). Next, the control centre sends an instruction equivalent to multiplying its current position by the complex number $z_2 = 3_{90^\circ}$. Calculate the polar coordinates of its final position and explain what geometric transformation it has undergone relative to its initial position vector.

Find the area of the triangle whose vertices are the affixes of the cube roots of the complex number $z = 64_{90^\circ}$.

Perform the following operation with complex numbers, expressing the final result in polar form:

To calculate the power $(-\sqrt{3} + i)^6$, two 1st-year Bachillerato students propose different approaches:

• Mateo proposes expanding it directly using Newton's binomial theorem: $\sum_{k=0}^6 \binom{6}{k} (-\sqrt{3})^{6-k} i^k$.

• Sofía proposes converting the base to polar form, applying De Moivre's formula, and expressing the final result in binomial form.

1. a) Reasonably compare both methods by analyzing the number of terms to calculate and the likelihood of making calculation errors (signs, powers of $i$, binomial coefficients).
2. b) Justify which method is optimal and execute it step by step to find the exact final result in polar form and in binomial form.
3. c) If the exponent were $2$ instead of $6$, would Sofía's method still be clearly superior? Justify your answer by comparing the algebraic effort in that case.

In a wind farm located in Albacete, a wind turbine has three blades of length $60\text{ m}$ symmetrically arranged around the central hub with an angular offset of $120^\circ$ from each other. We place the origin of coordinates at the center of the rotor and the angle $0^\circ$ on the right horizontal axis.

At an initial instant, the tip of blade 1 is located at position $z_1 = 60_{40^\circ}$ (in meters).

1. a) Model the position of the tips of blades 2 and 3 ($z_2$ and $z_3$) in polar form at that instant, explaining which operation with complex numbers in polar form allows moving from one blade to the next.
2. b) A constant gust of wind rotates the rotor $85^\circ$ counterclockwise. Model this rotation by multiplying by a unit complex number and calculate the new position in polar form of each of the three blades.
3. c) After the rotation, determine with reasoning which of the three blades is at the highest height relative to the ground (greatest vertical coordinate) and calculate that height relative to the rotor axis using trigonometric form.

An air traffic radar detects three aircraft whose positions form an equilateral triangle centred on the control tower (located at the origin of coordinates). If one of the aircraft is at the position described by the complex number $z_1 = 4 + 4\sqrt{3}i$ (in kilometres), determine the positions of the other two aircraft in polar form and calculate the straight-line distance separating any two aircraft.

Use de Moivre's formula to expand the power $(\cos \alpha + i \sin \alpha)^3$. From the result, by equating the real and imaginary parts, prove the triple-angle trigonometric identities:

• $\cos(3\alpha) = 4\cos^3 \alpha - 3\cos \alpha$

• $\sin(3\alpha) = 3\sin \alpha - 4\sin^3 \alpha$

Solve the equation $(z - 2i)^4 = 16_{180^\circ}$. Express the four solutions for $z$ in binomial form.

In the software design of a robotic arm, the final position of the end effector is calculated by applying a rotation and scaling to its initial position $z_0 = 5_{20^\circ}$. This transformation is equivalent to multiplying $z_0$ by the cube root of the complex number $W = 27_{150^\circ}$ whose argument is in the first quadrant. Determine the final position of the end effector in polar form and the total angle of rotation applied relative to the initial position.

Find two complex numbers $z_1$ and $z_2$ in polar form, knowing that they simultaneously satisfy the following conditions:

• Their quotient $\frac{z_1}{z_2}$ is equal to $2_{150^\circ}$

• Their product $z_1 \cdot z_2$ is equal to $18_{90^\circ}$

Calculate the sum of the six sixth roots of unity ($z^6 = 1$). Justify your answer in two different ways: algebraically (using the sum of a geometric progression or the properties of roots) and geometrically (interpreting the affixes as vectors).

An encryption system generates keys based on the coordinates of the affixes of the solutions to the equation $z^5 + 32 = 0$. Calculate all the solutions in polar form and determine the perimeter of the regular polygon they form.

Determine all complex numbers $z$ expressed in polar form that satisfy the equation $z^3 = (\bar{z})^2$, where $\bar{z}$ is the conjugate of $z$.

The solutions of a binomial equation of the form $z^n = w$ ($n \in \mathbb{N}$, $n \ge 3$, $w \in \mathbb{C}$) determine the vertices of a regular polygon centered at the origin of coordinates in the complex plane.

The following three conditions are known about this set of solutions:

1. One of the solutions is the complex number $z_1 = \sqrt{3} + i$.

2. Another of the solutions lies exactly on the positive imaginary semiaxis (its argument is $90^\circ$).

3. The regular polygon has the smallest possible number of sides ($n$) that simultaneously satisfies the two previous conditions.

1. a) Write $z_1$ in polar form and algebraically deduce the smallest integer value of $n$ compatible with one of the solutions having an argument of $90^\circ$.
2. b) Find the exact value of the complex number $w$ both in polar form and in binomial form.
3. c) Write all the solutions of the equation $z^n = w$ in polar form, calculating the angular step between consecutive vertices and checking that $z_1$ and the solution on the imaginary semiaxis are part of the list.

Consider the complex number $w = \dfrac{\sqrt{3}+i}{1-i}$.

1. a) Before performing analytic operations, make a reasoned estimate:

• Compare the moduli of the numerator and the denominator. From that comparison, deduce whether the affixes of the successive powers $w, w^2, w^3, \ldots, w^n$ will form a spiral moving away from the origin, one converging to the origin, or if they will remain on a fixed circle.

• Analyze the arguments of the numerator and the denominator to estimate whether it is possible for a power $w^n$ to exist (with $n \in \mathbb{N}$, $n \ge 1$) whose result is a strictly positive real number. Do you expect the smallest $n$ with that property to be a small value or a notable multiple?

1. b) Check your estimates:

• Convert the numerator and the denominator to polar form and obtain the exact expression of $w$ in polar form.

• Apply De Moivre's formula to determine exactly the smallest positive integer $n$ such that $w^n$ is a positive real number, stating and solving the condition on the argument.

1. c) For that minimum value of $n$, calculate the exact value of $w^n$ and of its inverse $(w^n)^{-1}$ in polar and binomial form, and evaluate whether the obtained modulus confirms your initial conjecture about the distance moving away from the origin.

Topic challenge. Design a technical report on the response of a grid-connected electrical installation using real data:

1. Look up the real-time mainland system demand (in GW) on the REE (Red Eléctrica de España) website and take the first two integer digits as the value $k$ (for example, if it reads $27{,}4$ GW, $k = 27$).

2. Model the circuit's response with the power polynomial $P(x) = \frac{1}{3}x^3 - \frac{k}{2}x^2 + 50x$. Find the derivative $P'(x)$, calculate the larger of the positive critical frequencies by solving $P'(x) = 0$, and round its value to two decimal places, calling it $\omega$.

3. Define the circuit impedance in rectangular (binomial) form as $Z = 4 + \omega i$ (in $\Omega$) and a load current $I = 3 - 2i$ (in A). Work in rectangular form to determine the complex voltage $V = Z \cdot I$.

4. Convert $Z$ and $I$ to polar form ($r_\alpha$). Compute the product in polar form to recalculate $V$, verify that both results match, and plot the three vector affixes in GeoGebra, analyzing the phase shift between current and voltage.

Submit a brief one-page report with your handwritten calculations and the GeoGebra screenshot.

Assessment criteria:

• Rigorous data extraction from REE and correct algebraic solution of the derivative $P'(x)$.

• Mastery and equivalence in complex number operations in both rectangular and polar forms.

• Graphical construction in GeoGebra and consistent physical interpretation of the phase shift.