Polar and Trigonometric Form of a Complex Number
A drone takes off from the Monumento de Santiago and moves 100 m in one direction. If you now want it to turn exactly 90° around the starting point, calculating the new coordinates $(x, y)$ with sines and cosines is heavy work. With complex numbers in polar form, rotating a point is just a multiplication. To get there, you need to describe each complex number by its distance from the origin and its angle.
Modulus and Argument: the Polar Coordinates of $z$
You already know how to place the point for $z=a+bi$ in the complex plane using the pair $(a, b)$. Another way to place it is to say how far from the origin it is and what angle it makes with the real axis.
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 Rectangular: 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 $b=r\,\sin\,\alpha$.
Move the point and watch how its coordinates change.
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.
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.
Conjugate, Opposite, and Equality in Polar Form
In rectangular form, the conjugate changes the sign of $b$ and the opposite changes both signs. In polar form you can see it in the angle: conjugating reflects the point across the real axis and the opposite rotates it half a turn.
When two polar forms are the same number
The modulus fixes the circle and the argument fixes the ray. Adding or subtracting full turns of $360^\circ$ does not move the point. That is why the argument is usually given between $0^\circ$ and $360^\circ$: $4_{-240^\circ}=4_{120^\circ}$ and $5_{30^\circ}=5_{390^\circ}$.
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 Inverse in Polar Form
Here is the great advantage of polar form. Multiply two numbers in trigonometric form and use the sine and cosine sum formulas:
$$r(\cos\alpha+i\,\sin\,\alpha)\cdot s(\cos\beta+i\,\sin\,\beta)=rs\big[(\cos\alpha\cos\beta-\sin\,\alpha\,\sin\,\beta)+i(\sin\,\alpha\cos\beta+\cos\alpha\,\sin\,\beta)\big]=rs\big[\cos(\alpha+\beta)+i\,\sin(\alpha+\beta)\big]$$Rotating a point
Multiplying by $1_\theta$ does not change the modulus and adds $\theta$ to the argument. That is, it rotates the point $\theta$ degrees counterclockwise around the origin. That is the answer to the drone at the beginning. To rotate the point $2+2i=2\sqrt2_{45^\circ}$ by $90^\circ$, you multiply by $1_{90^\circ}=i$ and get $2\sqrt2_{135^\circ}=-2+2i$.
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.
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.
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.
Powers: De Moivre's Formula
Raising to a power is multiplying the same number several times. The moduli are multiplied (raised to the $n$) and the arguments are added $n$ times (multiplied by $n$).
Trigonometric formulas with De Moivre
De Moivre also helps you discover identities. For $n=2$:
$(\cos\alpha+i\,\sin\,\alpha)^2=\cos2\alpha+i\,\sin\,2\alpha$.
If you expand the square, the left side is $(\cos^2\alpha-\sin^2\alpha)+i(2\,\sin\,\alpha\cos\alpha)$. Equating real parts and imaginary parts gives:
$\cos2\alpha=\cos^2\alpha-\sin^2\alpha$ and $\sin\,2\alpha=2\,\sin\,\alpha\cos\alpha$.
For $n=3$ you do the same with the cube of a binomial.
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.
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.
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 $\sin \alpha$.
$n$th Roots
Finding $\sqrt[n]{z}$ means finding the $w$ such that $w^n=z$. If $w=\rho_\varphi$ and $z=r_\alpha$, De Moivre says that $\rho^n=r$ and $n\varphi=\alpha+360^\circ k$. Solving:
This lets you solve equations like $z^4+81=0$: you write $z^4=-81=81_{180^\circ}$ and look for the fourth roots. If the equation has the form $z^6-7z^3-8=0$, a change of variable $u=z^3$ turns it into a quadratic in $u$. Then you find the cube roots of each value of $u$.
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.
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.
Summary
Go back to the beginning: to rotate the drone $90^\circ$ around the origin you multiply its position by $1_{90^\circ}=i$, because when multiplying the arguments add and the modulus does not change.