Complex Numbers in Trigonometric Form

4 min read

Multiplying two complex numbers in the rectangular form a + b i forces you to FOIL and then collect real and imaginary parts. Rewrite the same numbers in trigonometric form and multiplication becomes a single move: stretch by one length and turn by one angle. That is the payoff of viewing a complex number as a point with a distance and a direction instead of two coordinates.

By the end of this page you can:

• Plot a complex number a + b i in the complex plane and read off its modulus and argument.
• Write a complex number in the trigonometric form r(cos theta + i sin theta).
• Multiply and divide two complex numbers in trigonometric form.

The floor. Every nonzero complex number equals r(cos theta + i sin theta), where r is its distance from the origin and theta is the angle to the positive real axis. To multiply, multiply the moduli and add the arguments. To divide, divide the moduli and subtract the arguments.

Quick Reference

Before You Start

• Plot points and use distance and angle in polar coordinates.
• Read sine and cosine of a standard angle off the unit circle.
• Add, subtract, and multiply complex numbers in the form a + b i.

The Idea

A complex number a + b i is a point in the complex plane: the real part a runs left to right, the imaginary part b runs up and down. That point also has a distance from the origin and a direction. The distance is the modulus r, written |z|, and the direction is the argument theta, measured from the positive real axis. Reading a = r cos theta and b = r sin theta off that right triangle rewrites z as r(cos theta + i sin theta).

The reason this form is worth the trouble is multiplication. When you multiply two complex numbers, their moduli multiply and their arguments add. The two pieces, length and angle, never mix: a modulus is always a length and an argument is always an angle.

Key Formulas

For z = a + b i with r = |z| and theta the argument,

\[ z = r(\cos\theta + i\sin\theta), \qquad r = \sqrt{a^2 + b^2}, \qquad \tan\theta = \frac{b}{a}. \]

Source: Stewart 7e, Section 8.3 Polar Form of Complex Numbers; De Moivre’s Theorem.

For two numbers z_1 = r_1(cos theta_1 + i sin theta_1) and z_2 = r_2(cos theta_2 + i sin theta_2),

\[ z_1 z_2 = r_1 r_2\big(\cos(\theta_1+\theta_2) + i\sin(\theta_1+\theta_2)\big), \] \[ \frac{z_1}{z_2} = \frac{r_1}{r_2}\big(\cos(\theta_1-\theta_2) + i\sin(\theta_1-\theta_2)\big), \quad z_2 \ne 0. \]

Source: Stewart 7e, Section 8.3 Polar Form of Complex Numbers; De Moivre’s Theorem.

When you solve tan theta = b/a, choose theta in the quadrant where the point a + b i actually sits. The tangent equation alone has two answers half a turn apart, so the sign of a and the sign of b decide which one is correct.

Worked Example

Write z = 1 + i√3 in trigonometric form.

1. Find the modulus: r = √(a² + b²) = √(1² + (√3)²) = √(1 + 3) = √4 = 2.
2. Find a candidate angle: tan theta = b/a = √3 / 1 = √3.
3. Place the point: a = 1 > 0 and b = √3 > 0, so z is in the first quadrant. The first-quadrant angle with tangent √3 is theta = π/3 (60 degrees).
4. Assemble the form.

Connections

This builds on polar coordinates, since the trigonometric form is just the polar form of a point in the complex plane, and on unit-circle sine and cosine, which supply the values cos theta and sin theta. It leads into De Moivre’s theorem, which raises a complex number to a power and finds its roots by working with the modulus and argument directly.

Resources

• Stewart, Redlin, Watson, Precalculus 7e, Section 8.3 (primary).
• OpenStax Precalculus 2e, Section 10.5 Polar Form of Complex Numbers (alternate).

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Last updated: 2026-06-24