Polar and Trigonometric Form of a Complex Number

Modulus and Argument

A point in the plane can be described by two distances, $(a, b)$, or by one distance and one angle. That second description is called polar coordinates. Apply it to the point that represents $z = a + bi$ in the complex plane. The distance and the angle get special names.

A note on notation: you may also see $r\,\mathrm{cis}\,\theta$ (short for $r(\cos\theta + i\sin\theta)$), or a subscript, as in $4_{120^\circ}$. Some practice problems use the subscript form. It means exactly the same as $4\angle 120^\circ$. On this page we write $r\angle\theta$.

To go from $a + bi$ to polar form, find $r$ first. Then find the angle with a reference angle, $\tan^{-1}\left|\dfrac{b}{a}\right|$, and place it in the correct quadrant.

If $a = 0$, don't divide: $\theta = 90^\circ$ when $b > 0$ and $\theta = 270^\circ$ when $b < 0$. If $a = b = 0$, then $r = 0$ and there is no argument.

From Polar to Rectangular: Trigonometric Form

If you know $r$ and $\theta$, you can recover $a$ and $b$ with right-triangle trigonometry. The adjacent leg is $a = r\cos\theta$ and the opposite leg is $b = r\sin\theta$. Some problems call the form $a + bi$ Cartesian form. We will call it rectangular form.

Drag the point and watch the quadrant and the signs of $a$ and $b$. These exact values will help with the conversions:

For other quadrants, use the reference angle and fix the signs.

Conjugate, Opposite, and Equal Angles

In rectangular form, the conjugate flips the sign of $b$, and the opposite flips the signs of both $a$ and $b$. In polar form you can see both changes in the angle. The conjugate reflects the point across the real axis. The opposite turns it a half-turn.

Angles are only defined up to full turns, so two polar forms name the same number when the modulus matches and the angles differ by a multiple of $360^\circ$.

Multiplying and Dividing in Polar Form

This is where polar form pays off. Rectangular multiplication needs FOIL, and division needs the conjugate. In polar form you multiply the moduli and add the angles.

Why does this work? Multiply the trigonometric forms and use the angle-addition formulas:

The first bracket is $\cos(\theta_1 + \theta_2)$ and the second is $\sin(\theta_1 + \theta_2)$.

Rotations, Reciprocals, and Applications

The product rule has a geometric meaning. Multiplying by $w = s\angle\varphi$ scales the vector by $s$ and rotates it $\varphi$ counterclockwise. If $s = 1$, there is no scaling at all, so multiplying by $1\angle\varphi$ is a pure rotation by $\varphi$ about the origin. For example, $i = 1\angle 90^\circ$, so multiplying by $i$ is a quarter-turn.

The reciprocal (multiplicative inverse) of $z = r\angle\theta$ must satisfy $z \cdot \dfrac1z = 1 = 1\angle 0^\circ$. So the modulus is inverted and the angle is negated:

Engineers use this all the time. In an alternating-current circuit, the voltage $V$ and the impedance $Z$ are complex numbers, and the current is $I = V/Z$. You divide the moduli and subtract the angles. In a video game, a rotation of a character's position is just a multiplication by a unit complex number.

Powers and de Moivre's Theorem

A power is just repeated multiplication: each factor multiplies the modulus by $r$ and adds $\theta$ to the angle.

Each power turns another $40^\circ$ and, because $r > 1$, gets farther from the origin.

Back to the start: $1 + i = \sqrt2\angle 45^\circ$, so $(1+i)^8 = (\sqrt2)^8\angle(8 \cdot 45^\circ) = 16\angle 360^\circ = 16$. Eight turns of $45^\circ$ make one full circle, and the modulus grows to $16$.

More Uses of de Moivre's Theorem

De Moivre's theorem does more than compute powers. It also produces trigonometric identities and lets you handle huge exponents.

Identities. Take $r = 1$. Then $(\cos\alpha + i\sin\alpha)^n = \cos n\alpha + i\sin n\alpha$. Expand the left side with the binomial theorem and compare the real and imaginary parts of both sides.

Big exponents. If $|z| = 1$, the powers of $z$ just run around the unit circle and repeat. You only need the remainder of the exponent.

Roots of a Complex Number

Taking an $n$th root is de Moivre's theorem run backward. If $w = s\angle\varphi$ and $w^n = r\angle\theta$, then $s^n = r$ and $n\varphi = \theta + 360^\circ k$ for some integer $k$. Solve for $s$ and $\varphi$.

Why only $k = 0, \ldots, n-1$? Going from $k$ to $k + n$ adds $\dfrac{360^\circ n}{n} = 360^\circ$ to the angle, which is a full turn, so the roots start repeating.

Solving Equations with Powers and Roots

Two ideas combine to solve many equations. Use a substitution to make the equation look like $z^n = w$. Then use the root formula.

You can also find two unknown numbers from a quotient and a product. Multiply the two given equations to get $z_1^2$, then take a square root.

Roots as Polygons

The roots of $z^n = W$ are the vertices of a regular $n$-gon centered at the origin. That gives a geometric way to answer questions about perimeter, area and sums.

Why is the sum zero? Let $S$ be the sum of the roots and let $\omega = 1\angle\dfrac{360^\circ}{n}$. Multiplying every root by $\omega$ rotates the polygon onto itself, so it just shuffles the roots. Hence $\omega S = S$. Since $\omega \ne 1$, this forces $S = 0$.

Adding Complex Numbers and Conditions on the Argument

Polar form is great for products and powers, but it has no nice rule for sums. To add two numbers given in polar form, convert both to rectangular form, add the real and imaginary parts, and convert back if needed.

The same idea helps when a problem puts a condition on the argument. Put the number in the form $a + bi$ first. Then use the fact that $\tan\theta = \dfrac{b}{a}$, and check that $a$ and $b$ have the signs that match the quadrant.

Summary