Polar and Trigonometric Form of a Complex Number
A 2D video game moves a character by rotating it around a point. How can you rotate a point $60^\circ$ with a single multiplication? Until now you located $z$ with two Cartesian coordinates $(a, b)$. But a point can also be described by its distance from the origin and its angle, and with that information the product, quotient, powers, and roots become very simple. At the end of the page we answer the rotation question.
Modulus and argument
Since $a=r\cos\alpha$ and $b=r\sin\alpha$, the modulus comes from the Pythagorean theorem and the angle satisfies $\tan\alpha = \dfrac{b}{a}$.
Express the complex number $z = 3 + 3\sqrt{3}i$ in polar form and in trigonometric form.
Express the complex number $z = 3 + 3\sqrt{3}i$ in polar form and in trigonometric 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…
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.
From polar to binomial: trigonometric form
If you already have $r$ and $\alpha$, recovering $a$ and $b$ is right-triangle trigonometry: the adjacent leg is $a = r\cos\alpha$ and the opposite leg is $b = r\sin\alpha$.
Given the complex number in polar form $z = 4_{120^\circ}$, determine its expression in Cartesian form.
Given the complex number in polar form $z = 4_{120^\circ}$, determine its expression in Cartesian form.
Conjugate and opposite in polar form
In binomial form, conjugating changes the sign of $b$ and the opposite changes both signs. In polar form you see it in the angle: conjugating reflects across the real axis and the opposite rotates a half turn.
Negative angles or angles greater than $360^\circ$ are brought into the interval $[0^\circ, 360^\circ)$ by adding or subtracting full turns.
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…
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 $360^\circ$.
Product, quotient, and powers
This is where polar form shines. When you multiply $r(\cos\alpha+i\sin\alpha)$ by $s(\cos\beta+i\sin\beta)$ and use the sine and cosine sum formulas, the moduli are multiplied and the angles are added.
Multiplying by $w=1_{60^\circ}$ does not change the distance from the origin and adds $60^\circ$ to the angle: it is a rotation of $60^\circ$. That answers the video game question.
Calculate the product of the complex numbers $z_1 = 2_{30^\circ}$ and $z_2 = 5_{45^\circ}$. Express the result in polar 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.
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.
Apply De Moivre's formula to calculate the fourth power of the complex number $z = 2_{20^\circ}$. Express the result in polar 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…
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…
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.
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.
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 $z_1 = 3_{45^\circ}$…
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.
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.
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 $z_1 \times z_2 = 18_{60^\circ}$.…
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.
Roots of a complex number
If $w^n=z$, by De Moivre $w$ has modulus $\sqrt[n]{r}$ and angle satisfying $n\cdot\arg(w)=\alpha+360^\circ k$. Each value of $k$ gives a different angle until they start repeating.
This also solves equations like $z^4+81=0$: write $z^4=-81=81_{180^\circ}$ and find its four fourth roots.
Calculate the square roots of the complex number $z = 9_{80^\circ}$ and express the results in polar form.
Calculate the square roots of the complex number $z = 9_{80^\circ}$ and express the results in polar 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…
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.
Solve the equation $z^4 + 81 = 0$ using the polar form of complex numbers. Express the four solutions obtained in Cartesian form.
Solve the equation $z^4 + 81 = 0$ using the polar form of complex numbers. Express the four solutions obtained in Cartesian form.
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…
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.
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 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).
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…
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.
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.
Combined problems
Sometimes it is convenient to use binomial form to add and polar form to multiply. To add vectors, convert each one to binomial form, add the real parts and the imaginary parts, and if necessary go back to polar.
Determine the value of $a \in \mathbb{R}$ such that the argument of the complex number $z = \frac{a + 2i}{1 + i}$ is $45^\circ$.
Determine the value of $a \in \mathbb{R}$ such that the argument of the complex number $z = \frac{a + 2i}{1 + i}$ is $45^\circ$.
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, $z_2$,…
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…
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 $\sin \alpha$.
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…
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 $S = z^{2023} + z^{2024} + z^{2025}$.