Exercises: Polar and trigonometric form of a complex number

Express the complex number $z = 3 + 3\sqrt{3}i$ in polar form and in trigonometric form.

Given the complex number in polar form $z = 4_{120^\circ}$, determine its expression in Cartesian form.

Calculate the product of the complex numbers $z_1 = 2_{30^\circ}$ and $z_2 = 5_{45^\circ}$. Express the result in polar form.

Divide the complex numbers $z_1 = 12_{150^\circ}$ and $z_2 = 3_{60^\circ}$. Express the final result in trigonometric form.

Apply De Moivre's formula to calculate the fourth power of the complex number $z = 2_{20^\circ}$. Express the result in polar form.

Find the conjugate and the opposite of the complex number $z = 6_{45^\circ}$. Express both results in polar form with positive arguments between $0^\circ$ and $360^\circ$.

Calculate the square roots of the complex number $z = 9_{80^\circ}$ and express the results in polar form.

Observe the representation in the complex plane of the vector defining $z$. Determine its modulus and its argument, and write its expression in polar and trigonometric form.

Seán has to convert the complex number $z = -2 + 2\sqrt{3}i$ to polar form. On his exam, he writes:

"$r = \sqrt{(-2)^2 + (2\sqrt{3})^2} = \sqrt{4 + 12} = \sqrt{16} = 4$.

$\alpha = \arctan\left(\frac{2\sqrt{3}}{-2}\right) = \arctan(-\sqrt{3}) = -60^\circ = 300^\circ$.

Therefore, $z = 4_{300^\circ}$".

Identify the conceptual error Seán made when determining the argument, explain by analysing the signs of the real part and the imaginary part which quadrant the affix of $z$ must lie in, and write the correct polar form with its principal argument in the interval $[0^\circ, 360^\circ)$.

Find a complex number $z$ whose affix lies in the fourth quadrant and whose distance from the origin of coordinates is equal to $6$. It is known that its conjugate, $\bar{z}$, satisfies that its real part is exactly half of its modulus ($a = r/2$). Justify your reasoning to determine the argument of $\bar{z}$, deduce the polar form of $z$ with its argument in the interval $[0^\circ, 360^\circ)$, and find the exact binomial form of its opposite, $-z$.

In an alternating current circuit, the voltage is represented by the complex number $V = 220_{30^\circ}$ and the impedance by $Z = 10_{-15^\circ}$. Ohm's law for alternating current states that the current is $I = \frac{V}{Z}$. Calculate the current $I$ in polar form and subsequently express it in Cartesian form.

In 2D video game development, the position of a character is given by the affix of the complex number $z = 4 - 4i$. The graphics engine applies an anticlockwise rotation of $60^\circ$ by multiplying its position by the complex number $w = 1_{60^\circ}$. Determine the new position of the character in Cartesian form.

Find the three cube roots of the complex number $z = 8i$. Express the results in polar form and justify geometrically the shape formed by their affixes in the complex plane.

Calculate the value of the expression $\frac{(1 - \sqrt{3}i)^5}{(-2+2i)^3}$. To do this, first convert the numerator and denominator to polar form.

In an optics experiment, the phase difference between two light waves is modelled by multiplying their complex representations. If the first wave is $z_1 = 3_{45^\circ}$ and the second is $z_2 = 2_{135^\circ}$, find the result of $z_1 \times z_2^2$ in trigonometric form.

Find the reciprocal of the complex number $z = 4_{210^\circ}$ and express it in Cartesian form. Check that the product of $z$ and its reciprocal results in the real unit.

Solve the equation $z^4 + 81 = 0$ using the polar form of complex numbers. Express the four solutions obtained in Cartesian form.

Determine the value of $a \in \mathbb{R}$ such that the argument of the complex number $z = \frac{a + 2i}{1 + i}$ is $45^\circ$.

Consider the complex number $z = -3\sqrt{2} - \sqrt{6}i$.

1. a) Before calculating: knowing that $\sqrt{2} \approx 1.41$ and $\sqrt{6} \approx 2.45$, reasonably estimate the quadrant in which its affix lies. Compare the absolute value of the real part ($|a| \approx 4.24$) against that of the imaginary part ($|b| \approx 2.45$): will the principal argument be closer to $180^\circ$ or to $270^\circ$? Provide an estimate of its argument (within a margin of $\pm 10^\circ$) and an approximation of its modulus.
2. b) Algebraically calculate the exact value of the modulus $r$ and the principal argument $\alpha$ of $z$. Write $z$ in polar form and in trigonometric form, and compare the exact result with the estimates made in the previous section, justifying whether the calculation is consistent with the geometric position of the affix.

To calculate the algebraic (binomial) form of the opposite of the conjugate of the complex number $z = 4_{150^\circ}$, that is, $-\bar{z}$, two students debate which strategy is more convenient:

• Method 1 (binomial approach): first convert $z = 4_{150^\circ}$ to algebraic form using $a = r\cos\alpha$ and $b = r\sin\alpha$, then calculate its conjugate $\bar{z} = a - bi$, and finally change the sign of both components to obtain $-\bar{z}$.

• Method 2 (polar approach): directly calculate the argument of $-\bar{z}$ using the properties of the argument of the conjugate ($\arg(\bar{z}) = -\alpha$) and of the opposite ($\arg(-w) = \arg(w) + 180^\circ$), simplify the resulting angle in $[0^\circ, 360^\circ)$ knowing that the modulus does not change, and only convert the final result to algebraic form.

Apply both methods step by step to find $-\bar{z}$ in algebraic form and argue which of the two approaches minimizes the risk of making calculation errors with signs and trigonometric ratios.

The vertices of a regular hexagon centred at the origin of the complex plane correspond to the sixth roots of a complex number $W$. If one of the vertices of the hexagon is the affix of the number $z_1 = 2_{30^\circ}$, determine the value of the complex number $W$ in Cartesian form and calculate the exact area of the hexagon.

An articulated robotic arm consists of two segments. The first segment is represented in the complex plane by the vector $z_1 = 5_{60^\circ}$. The second segment, $z_2$, has a length of $3$ units and is rotated by $45^\circ$ anticlockwise with respect to the direction of $z_1$. Calculate the final position of the end of the robotic arm (that is, $z_1 + z_2$) in Cartesian form, rounding to two decimal places.

You can do it on paper and send a photo with your answer.

Use De Moivre's formula to expand the expression $(\cos \alpha + i \sin \alpha)^3$. From the equality between the real and imaginary parts, deduce the trigonometric identities for $\cos(3\alpha)$ and $\sin(3\alpha)$ in terms of $\cos \alpha$ and $\sin \alpha$.

Find all complex numbers $z$ that satisfy the equation $z^6 - 7z^3 - 8 = 0$. Express the solutions in polar form. (Hint: use a change of variable).

Find two complex numbers $z_1$ and $z_2$ knowing that their quotient is $\frac{z_1}{z_2} = 2_{120^\circ}$ and that their product is $z_1 \times z_2 = 18_{60^\circ}$. Express both numbers in polar form.

Five coplanar forces of equal magnitude $F = 10\text{ N}$ are applied to a point mass located at the origin. The directions of these forces correspond to the affixes of the fifth roots of the complex number $z = 100000_{0^\circ}$. Prove mathematically, using the properties of complex numbers in polar form, that the resultant force is zero.

Calculate the perimeter of the polygon formed by the affixes of the solutions to the equation $z^4 + 2z^2 + 4 = 0$ in the complex plane.

Let $z = -\frac{1}{2} + \frac{\sqrt{3}}{2}i$ be the complex number. Prove that $z^3 = 1$ and then calculate the value of the expression $S = z^{2023} + z^{2024} + z^{2025}$.

An oceanographic buoy from the Marine Institute anchored off the coast of Galway records the combined action of wind and ocean currents for Met Éireann forecast models. The force on the sensor is modeled in the complex plane, where the positive real semi-axis points East and the positive imaginary semi-axis points North (forces are expressed in kilonewtons, $\text{kN}$).

The sensor detects a force due to the current of $F_1 = 3\sqrt{3} - 3i\text{ kN}$. Simultaneously, a gust of wind introduces a second force $F_2$ whose modulus is $6\text{ kN}$ and whose affix forms an angle of $150^\circ$ counterclockwise with respect to the current vector $F_1$.

1. a) Determine the modulus and the principal argument of $F_1$ to express this force in polar form, and indicate in which geographic quadrant the ocean current acts.
2. b) Deduce the polar form of the wind force $F_2$ in the general system of the complex plane and find its exact algebraic (binomial) form.
3. c) Calculate the net force $F_{\text{total}} = F_1 + F_2$ in algebraic form, express it in polar form, and physically interpret the result: with what total intensity and in which net geographic direction does the water push the buoy?

Consider two non-zero complex numbers, $z_1 = r_\alpha$ and $z_2 = s_\beta$, with principal arguments expressed in the interval $[0^\circ, 360^\circ)$. The following information is known about them:

1. The product of their moduli is $r \times s = 12$ and the quotient of their moduli $r / s$ is equal to $3$.

2. The quotient $z_1 / z_2$ is a pure imaginary number with a strictly negative imaginary part.

3. The affix of the opposite of $z_2$, that is, $-z_2$, lies on the angle bisector of the fourth quadrant.

1. a) From the properties of the modulus and the geometric position of the opposite, determine the exact value of the modulus $s$ and the argument $\beta$ of $z_2$.
2. b) Using the condition on the quotient $z_1 / z_2$, determine the modulus $r$ and all possible principal arguments $\alpha \in [0^\circ, 360^\circ)$ of $z_1$.
3. c) Write the solutions found for $z_1$ and $z_2$ in exact trigonometric form and analytically verify that the quotient $z_1 / z_2$ satisfies all the given conditions.