Trigonometric Form of Complex Numbers — Mathematics Knowledge System

Overview

Every complex number z = a + bi can be located as a point in the complex plane and described not just by its coordinates (a, b) but by its distance from the origin (the modulus r) and the angle its position vector makes with the positive real axis (the argument θ). Rewriting z in terms of r and θ instead of a and b gives the trigonometric (polar) form z = r(cosθ + i sinθ), which turns multiplication and division of complex numbers — normally messy FOIL-and-simplify arithmetic — into simple rules of adding or subtracting angles and multiplying or dividing moduli. This is the bridge between complex-number algebra and trigonometry, and it is the form that makes De Moivre's theorem and roots of complex numbers tractable.

Intuition

Plot z = a + bi as the point (a, b). The segment from the origin to that point has length r (found by the Pythagorean theorem) and makes angle θ with the positive real axis, exactly like locating a point by polar coordinates. Since a = r cosθ and b = r sinθ come straight from right-triangle trigonometry, substituting them back into a + bi produces the trigonometric form automatically. The payoff is that multiplying two complex numbers in this form stretches (multiplies moduli) and rotates (adds arguments) — geometry that is invisible in the rectangular a + bi form but obvious once you see z as an arrow with a length and a direction.

Formal Definition

Definition

For a nonzero complex number z = a + bi, let r = |z| be its modulus and θ = arg(z) be its argument (angle from the positive real axis, typically taken in (−180°, 180°] or [0°, 360°)):

r = |z| = \sqrt{a^2+b^2}

Modulus

\cos\theta = \frac{a}{r}, \quad \sin\theta = \frac{b}{r}, \quad \tan\theta = \frac{b}{a}\ (a \ne 0)

Argument via the reference right triangle

z = a+bi = r(\cos\theta + i\sin\theta)

Trigonometric (polar) form

z_1 z_2 = r_1 r_2\big(\cos(\theta_1+\theta_2) + i\sin(\theta_1+\theta_2)\big)

Product rule: multiply moduli, add arguments

\frac{z_1}{z_2} = \frac{r_1}{r_2}\big(\cos(\theta_1-\theta_2) + i\sin(\theta_1-\theta_2)\big)

Quotient rule: divide moduli, subtract arguments

Worked Examples

Compute the modulus.
$r = \sqrt{3^2 + (3\sqrt3)^2} = \sqrt{9+27} = \sqrt{36} = 6$
Compute the argument using tanθ = b/a.
$\tan\theta = \frac{3\sqrt3}{3} = \sqrt3 \;\Rightarrow\; \theta = 60^\circ \text{ (since } a,b>0\text{, first quadrant)}$
Assemble the trigonometric form.
$z = 6(\cos 60^\circ + i\sin 60^\circ)$

Answer: z = 6(cos 60° + i sin 60°)

Practice Problems

Difficulty 4/10

Write z = 1 − i in trigonometric form.

Difficulty 5/10

Divide z₁ = 12(cos 150° + i sin 150°) by z₂ = 3(cos 30° + i sin 30°) using the trigonometric-form quotient rule.

Difficulty 6/10

In AC circuit analysis, an impedance is represented as a complex number with modulus 10 ohms and argument 36.87° (since tan(36.87°) ≈ 0.75). Write this impedance in trigonometric form and find its rectangular (a + bi) form, given cos(36.87°) ≈ 0.8 and sin(36.87°) ≈ 0.6.

Quiz

Summary

Trigonometric form: z = r(cosθ + isinθ), with r = √(a²+b²) and θ = arg(z) from tanθ = b/a (quadrant-adjusted).
Multiplying complex numbers in this form multiplies moduli and adds arguments; dividing divides moduli and subtracts arguments.
This form converts complex-number arithmetic into a scale-and-rotate geometric operation in the complex plane.
It is the essential setup for De Moivre's theorem and for computing powers and roots of complex numbers.

References

WebsiteWikipedia — Complex number § Polar form