Operations with Complex Numbers in Polar Form
A drone turns $90^\circ$ and moves away to twice the distance. In binomial form, calculating its new position requires expanding products. In polar form, you just multiply two numbers and add two angles. That is why polar form is used when you need to raise to a high power or take roots.
A note on notation: here we write $r\angle\theta$ for the complex number with modulus $r$ and argument $\theta$. Some problems write it with a subscript, $r_\theta$, and it means the same thing.
Product and quotient: moduli and arguments separately
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.
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:
$$ \frac{2_{20^\circ} \times 6_{70^\circ}}{4_{40^\circ}} $$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?
Conjugate and opposite in polar form
If $z=r\angle\alpha$, the conjugate $\bar z$ is its reflection across the real axis: same modulus and opposite angle. The opposite $-z$ is its reflection through the origin: same modulus and an angle $180^\circ$ larger.
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.
Adding complex numbers: converting from polar to binomial and back
Product and quotient are easy in polar form, but addition is not: you can't add moduli or arguments. To add or subtract, convert to binomial form $a+bi$, add the real and imaginary parts, and, if needed, convert back to polar.
This is useful in electricity: in a parallel circuit, $\dfrac1{Z_T}=\dfrac1{Z_1}+\dfrac1{Z_2}$. Each impedance is inverted in polar form, converted to binomial, added, and inverted again.
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.
De Moivre's formula: powers in polar form
Raising to a power is repeated multiplication. Applying the product rule $n$ times, the modulus is multiplied by itself $n$ times ($r^n$) and the argument is added to itself $n$ times ($n\alpha$).
De Moivre to obtain trigonometric identities
De Moivre's formula gives two expressions for the same number: one by expanding the binomial and another with the multiplied angle. Two complex numbers are equal if their real parts and their imaginary parts coincide, so equating them yields identities.
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.
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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$\sin(3\alpha) = 3\sin \alpha - 4\sin^3 \alpha$
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.
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, $\pm2$. Among the complex numbers, $z^n = w$ has exactly $n$ solutions, and their affixes are the vertices of a regular $n$-sided polygon centered at the origin.
Equations with a shift: $(z-a)^n=w$
If the equation has the form $(z-a)^n=w$, let $u=z-a$, solve $u^n=w$ with the nth roots, and at the end solve for $z=a+u$. Geometrically, the polygon of the roots is translated: its center is no longer the origin but the point $a$.
Side, perimeter, and area of the polygon
Two consecutive vertices and the origin form an isosceles triangle: two sides measure $R$ (the modulus of the roots) and the angle between them is $\frac{360^\circ}{n}$. If you draw the altitude, the triangle splits into two right triangles with angle $\frac{180^\circ}{n}$ and hypotenuse $R$. Half the side is the opposite leg: $R\operatorname{sen}\frac{180^\circ}{n}$.
For the area, the polygon is divided into $n$ equal triangles with a vertex at the origin. Each one has area $\frac12R\cdot R\operatorname{sen}\frac{360^\circ}{n}$ (two sides and the angle between them).
This same side formula works for the distance between two airplanes at the vertices of an equilateral triangle ($n=3$): $\ell=2R\operatorname{sen}60^\circ=R\sqrt3$.
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).
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.
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.
Equations with complex numbers: combining the rules
The hard problems mix everything above. The strategy is to equate moduli and arguments separately, remembering that two arguments are equal if they differ by a multiple of $360^\circ$.
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 $z$.
Back to the beginning: turning $90^\circ$ and doubling the distance is equivalent to multiplying by $2\angle90^\circ$, that is, adding $90^\circ$ to the argument and doubling the modulus.