Operations with complex numbers in polar and trigonometric form
An inspection drone flying over rice fields takes off from the origin and reaches a point. If the pilot wants it to rotate 90° around the origin without getting closer or farther away, how many calculations does the pilot need? In binomial form, multiplying two complex numbers forces you to expand four products. In polar form, you just multiply two numbers and add two angles. That is why polar form is the one used for high powers and roots.
Recall the notation: $z=r_\theta$ is the same as $z=r(\cos\theta+i\,\mathrm{sin}\,\theta)$, with modulus $r$ and argument $\theta$.
Product and quotient: moduli and arguments separately
Conjugate and opposite
The conjugate of $r_\alpha$ is $r_{-\alpha}$ (symmetric about the real axis). The opposite is $r_{\alpha+180^\circ}$ (same modulus, half a turn). For example, for $z=7_{100^\circ}$: $\bar z=7_{-100^\circ}=7_{260^\circ}$ and $-z=7_{280^\circ}$.
Given the complex numbers $z_1 = 4_{30^\circ}$ and $z_2 = 3_{60^\circ}$, calculate their product $z_1 \times z_2$, expressing the result in polar form.
Given the complex numbers $z_1 = 4_{30^\circ}$ and $z_2 = 3_{60^\circ}$, calculate their product $z_1 \times z_2$, expressing the result in polar form.
Calculate the quotient $\frac{12_{150^\circ}}{3_{45^\circ}}$ and express the result in polar form.
Calculate the quotient $\frac{12_{150^\circ}}{3_{45^\circ}}$ 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.
Given the complex number $z = 3_{50^\circ}$, express its conjugate $\bar{z}$ and its opposite $-z$ in polar form.
De Moivre's formula: powers
Raising to a power is repeated multiplication. Applying the product rule $n$ times, the modulus becomes $r^n$ and the argument becomes $n\alpha$.
Using de Moivre's formula, calculate the power $(2_{40^\circ})^3$ 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.
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.
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.
Determine the smallest positive integer $n$ such that the result of $(\sqrt{3} + i)^n$ is a negative real number.
The inverse: inverse modulus, opposite argument
Find the multiplicative inverse of the complex number $z = 5_{120^\circ}$, expressing the final 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…
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?
Perform the following combined operation and express the result in polar form: $$ \frac{2_{20^\circ} \times 6_{70^\circ}}{4_{40^\circ}} $$
Perform the following combined operation and express the result in polar form:
$$ \frac{2_{20^\circ} \times 6_{70^\circ}}{4_{40^\circ}} $$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 $z_1 = 2_{45^\circ}$…
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 kilometers). Next, the control center 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.
Perform the following operation with complex numbers, expressing the final result in polar form: $$ \frac{(2_{30^\circ})^4 \times \sqrt{3}{150^\circ}}{4_{60^\circ}} $$
Perform the following operation with complex numbers, expressing the final result in polar form:
$$ \frac{(2_{30^\circ})^4 \times \sqrt{3}_{150^\circ}}{4_{60^\circ}} $$nth roots: $n$ solutions on a regular polygon
The equation $x^2=4$ has two real solutions, $\pm2$. In the complex numbers, $z^n=w$ (with $w\neq0$) has exactly $n$ solutions, and their points in the plane form the vertices of a regular $n$-sided polygon centered at the origin.
Calculate the two square roots of the complex number $16_{120^\circ}$ and express the results in polar form.
Calculate the two square roots of the complex number $16_{120^\circ}$ and express the results in polar form.
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…
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.
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…
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).
Sides, perimeter and area of the polygon of the roots
The points in the complex plane that represent the roots are the vertices of a regular polygon inscribed in a circle of radius $R=\sqrt[n]{s}$. To measure it you need the distance between two points.
For the area of a triangle with vertices $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$ you can use $A=\dfrac12\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|$. If it is equilateral with side $\ell$, $A=\dfrac{\sqrt3}{4}\ell^2$ is enough.
Find the area of the triangle whose vertices are the affixes of the cube roots of the complex number $z = 64_{90^\circ}$.
Find the area of the triangle whose vertices are the affixes of the cube roots of the complex number $z = 64_{90^\circ}$.
An air traffic radar detects three aircraft whose positions form an equilateral triangle centered on the control tower (located at the origin of coordinates). If one of…
An air traffic radar detects three aircraft whose positions form an equilateral triangle centered 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 kilometers), determine the positions of the other two aircraft in polar form and calculate the straight-line distance separating any two aircraft.
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…
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.
Combined problems
The hardest problems mix the previous rules. Strategy: (1) convert everything to polar form, (2) translate the conditions into equations for moduli and arguments, (3) solve, and (4) check.
In an alternating current circuit, the total impedance $Z_T$ of two components connected in parallel is calculated using the formula…
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.
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…
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:
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$\cos(3\alpha) = 4\cos^3 \alpha - 3\cos \alpha$
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$\sin(3\alpha) = 3\sin \alpha - 4\sin^3 \alpha$
Solve the equation $(z - 2i)^4 = 16_{180^\circ}$. Express the four solutions for $z$ in binomial form.
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…
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…
Find two complex numbers $z_1$ and $z_2$ in polar form, knowing that they simultaneously satisfy the following conditions:
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Their quotient $\frac{z_1}{z_2}$ is equal to $2_{150^\circ}$
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Their product $z_1 \times z_2$ 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…
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).
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 $z$.
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 $z$.