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 theorem, calculate the power $(2_{40^\circ})^3$ and express the result in polar form.

Find the 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 make a turn of $90^\circ$ counterclockwise 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.

Solve the equation $z^4 + 81 = 0$ in the set of complex numbers. Express the solutions in polar form and represent them graphically.

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 Cartesian 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 and Cartesian form.

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

Determine the smallest positive integer $n$ so 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 miles). Then, the control center sends it 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 with respect 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:

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

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

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

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

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

In the software design of a robotic arm, the final position of the effector is calculated by applying a rotation and a 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 $W = 27_{150^\circ}$ that has its argument in the first quadrant. Determine the final position of the effector in polar form and the total angle of rotation applied with respect 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 the complex numbers $z$ expressed in polar form that satisfy the equation $z^3 = (\bar{z})^2$, where $\bar{z}$ is the conjugate of $z$.