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.

Perform the division of 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 theorem 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 on the complex plane of the vector that defines $z$. Determine its modulus and its argument, and write its expression in polar and trigonometric form.

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 the development of a 2D video game, the position of a character is given by the point representing the complex number $z = 4 - 4i$. The graphics engine applies a counterclockwise 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 geometrically justify the figure formed by their points on 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 the denominator to polar form.

In an optics experiment, the phase difference between two light waves is modeled 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 $z_1 \cdot z_2^2$ in trigonometric form.

Find the inverse of the complex number $z = 4_{210^\circ}$ and express it in Cartesian form. Prove that the product of $z$ and its inverse 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 $a \in \mathbb{R}$ so that the argument of the complex number $z = \frac{a + 2i}{1 - i}$ is $45^\circ$.

The vertices of a regular hexagon centered 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 $45^\circ$ counterclockwise with respect to the direction of $z_1$. Calculate the final position of the end of the robotic arm (i.e., $z_1 + z_2$) in Cartesian form, rounding to two decimal places.

Use de Moivre's theorem 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 the complex numbers $z$ that satisfy the equation $z^6 - 7z^3 - 8 = 0$. Express the solutions in polar form. (Hint: use a substitution).

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

Five coplanar forces of equal magnitude $F = 10\text{ N}$ are applied to a material point 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 the complex number be $z = -\frac{1}{2} + \frac{\sqrt{3}}{2}i$. Prove that $z^3 = 1$ and then calculate the value of the expression $S = z^{2023} + z^{2024} + z^{2025}$.