Polar and trigonometric form of a complex number · Mathematics 6th Year HL

Modulus and argument: the polar coordinates of $z$

Practice★☆☆

Express the complex number $z = 3 + 3\sqrt{3}i$ in polar form and in trigonometric form.

Practice★☆☆

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.

Solve

Practice★☆☆

Seán has to convert the complex number $z = -2 + 2\sqrt{3}i$ to polar form. On his exam, he writes: "$r = \sqrt{(-2)^2 + (2\sqrt{3})^2} = \sqrt{4 + 12} = \sqrt{16} = 4$.…

Seán has to convert the complex number $z = -2 + 2\sqrt{3}i$ to polar form. On his exam, he writes:

"$r = \sqrt{(-2)^2 + (2\sqrt{3})^2} = \sqrt{4 + 12} = \sqrt{16} = 4$.

$\alpha = \arctan\left(\frac{2\sqrt{3}}{-2}\right) = \arctan(-\sqrt{3}) = -60^\circ = 300^\circ$.

Therefore, $z = 4_{300^\circ}$".

Identify the conceptual error Seán made when determining the argument, explain by analysing the signs of the real part and the imaginary part which quadrant the affix of $z$ must lie in, and write the correct polar form with its principal argument in the interval $[0^\circ, 360^\circ)$.

Solve

From polar to rectangular form

If you already have $r$ and $\alpha$, recovering $a$ and $b$ is the same trigonometry as in the right-angled triangle: the adjacent side is $a = r\cos\alpha$ and the opposite side is $b = r\sin\alpha$.

Practice★☆☆

Given the complex number in polar form $z = 4_{120^\circ}$, determine its expression in Cartesian form.

Solve

Conjugate and negative in polar form

In rectangular form, conjugating changes the sign of $b$ and the negative $-z$ changes both signs. In polar form the idea is the same, but it shows up in the angle: conjugating reflects the argument in the real axis, and the negative rotates it by $180^\circ$.

Practice★☆☆

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 $360^\circ$.

Solve

Solving equalities in polar form

Two complex numbers in polar form are equal if they have the same modulus and arguments that differ by a multiple of $360^\circ$

For $r,s>0$, $r(\cos\theta+i\sin\theta) = s(\cos\varphi+i\sin\varphi) \iff r=s$ and $\theta=\varphi+360^\circ k$ (that is, $2\pi k$ in radians) for some integer $k$. That is why the argument is given in a fixed range such as $(-180^\circ,180^\circ]$, so that each non-zero complex number has a single "standard" polar representation. With modulus zero, all the trigonometric expressions represent 0, whatever angle is chosen; that angle is not an argument of 0.

Why? The modulus fixes the radius of the circle on which the point lies, and the argument defines the direction of the half-line. Adding or subtracting multiples of $360^\circ$ amounts to going round complete turns, so the half-line is the same and the point representing $z$ does not change position.

Practice★★☆

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.

Solve

Multiplying and dividing in polar form

In rectangular form, multiplying two complex numbers means expanding brackets. In polar form it is just arithmetic on the moduli and the arguments.

Product, quotient and reciprocal

For $z_1 = r_1(\cos\theta_1 + i\sin\theta_1)$ and $z_2 = r_2(\cos\theta_2 + i\sin\theta_2)$: $z_1z_2 = r_1r_2\big(\cos(\theta_1+\theta_2) + i\sin(\theta_1+\theta_2)\big)$ and $\dfrac{z_1}{z_2} = \dfrac{r_1}{r_2}\big(\cos(\theta_1-\theta_2) + i\sin(\theta_1-\theta_2)\big)$. Multiply the moduli and add the arguments; divide the moduli and subtract the arguments. In particular $\dfrac{1}{z} = \dfrac{1}{r}\big(\cos(-\theta)+i\sin(-\theta)\big)$, and multiplying by a number of modulus 1 rotates $z$ anticlockwise about the origin by the argument of that number.

Why? Multiply out and use $\cos(A+B)$ and $\sin(A+B)$: $(\cos A + i\sin A)(\cos B + i\sin B) = (\cos A\cos B - \sin A\sin B) + i(\sin A\cos B + \cos A\sin B) = \cos(A+B) + i\sin(A+B)$. The quotient follows from $\dfrac{1}{z_2} = \dfrac{\bar z_2}{r_2^2}$, because $z\bar z = |z|^2$.

Practice★☆☆

Calculate the product of the complex numbers $z_1 = 2_{30^\circ}$ and $z_2 = 5_{45^\circ}$. Express the result in polar form.

Solve

Practice★☆☆

Divide the complex numbers $z_1 = 12_{150^\circ}$ and $z_2 = 3_{60^\circ}$. Express the final result in trigonometric form.

Solve

Practice★★☆

Determine the value of $a \in \mathbb{R}$ such that the argument of the complex number $z = \frac{a + 2i}{1 + i}$ is $45^\circ$.

Solve

Powers: De Moivre's Theorem

Repeating the product rule $n$ times gives a rule for powers.

If $z^n = 1$, the powers of $z$ repeat with period $n$, so you can reduce a large exponent: $z^{3k+1} = z$ when $z^3 = 1$.

Practice★☆☆

Apply De Moivre's formula to calculate the fourth power of the complex number $z = 2_{20^\circ}$. Express the result in polar form.

Solve

Practice★★☆

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.

Solve

Practice★★★

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 $\sin \alpha$.

Solve

Roots of complex numbers

To solve $z^n = a$, write $a = r(\cos\varphi + i\sin\varphi)$ and $z = \rho(\cos\theta + i\sin\theta)$. By De Moivre, $z^n = \rho^n(\cos n\theta + i\sin n\theta)$. Equality in polar form gives $\rho^n = r$ and $n\theta = \varphi + 360^\circ k$.

If the equation is $z^{2n} + bz^n + c = 0$, substitute $w = z^n$, solve the quadratic in $w$ (with the quadratic formula; the roots may be complex), then find the $n$th roots of each value of $w$.

The $n$th roots of unity add up to $0$ for $n > 1$: they are $1, w, w^2, \dots, w^{n-1}$ with $w = \cos\dfrac{360^\circ}{n} + i\sin\dfrac{360^\circ}{n}$, a geometric series with ratio $w \ne 1$ and $w^n = 1$, so the sum is $\dfrac{1 - w^n}{1 - w} = 0$. Geometrically, equally spaced vectors of equal length cancel out.

Practice★☆☆

Calculate the square roots of the complex number $z = 9_{80^\circ}$ and express the results in polar form.

Solve

Practice★★☆

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

Practice★★☆

Solve the equation $z^4 + 81 = 0$ using the polar form of complex numbers. Express the four solutions obtained in Cartesian form.

Solve

Back to the start: Seán's marker is $6(\cos60^\circ + i\sin60^\circ)$, which is 6 km away at 60° north of east, the same as $3 + 3\sqrt3\, i$ (3 km east and about 5.2 km north).

Summary

Modulus $|z|=\sqrt{a^2+b^2}$ (distance from the origin); argument $\arg(z)$ (angle with the positive real axis, taking care over the quadrant).
Polar form $z=r(\cos\theta+i\sin\theta)$: the Cartesian↔polar change of coordinates applied to the point on the Argand diagram.
Rectangular→polar: $r=\sqrt{a^2+b^2}$, $\theta=\arctan(b/a)$ correcting for the quadrant. Polar→rectangular: $a=r\cos\theta$, $b=r\sin\theta$.
Conjugate has argument $-\theta$; negative $-z$ has argument $\theta+180^\circ$: same modulus, argument reflected or rotated by a half-turn.
For positive moduli, two polar forms are the same number if they have the same modulus and arguments that differ by a multiple of $360^\circ$.
Multiply moduli and add arguments; divide moduli and subtract arguments.
De Moivre's Theorem: $[r(\cos\theta+i\sin\theta)]^n = r^n(\cos n\theta+i\sin n\theta)$. The $n$th roots of $a$ are $n$ points equally spaced on a circle of radius $r^{1/n}$.

Let's practice! 10 easy exercises in this section

★☆☆ Express the complex number $z = 3 + 3\sqrt{3}i$ in polar form and in trigonometric form.★☆☆ Given the complex number in polar form $z = 4_{120^\circ}$, determine its expression in…

Let's practice! 10 medium exercises in this section

★★☆ In an alternating current circuit, the voltage is represented by the complex number…