The Trigonometric (Polar) Form of a Complex Number
Simple Explanation
Instead of writing a complex number by its real and imaginary parts, it can be written using its modulus r and its argument θ (the angle from the positive real axis): z = r(cos θ + i sin θ).
Why Do We Need It?
Trigonometric form makes multiplying, dividing, and raising complex numbers to powers dramatically simpler than working directly with a+bi — this is the form De Moivre's Theorem is built on.
See It
A point representing z=3+4i on the complex plane, with the angle theta marked between the positive real axis and the line to z
Formula
The Trigonometric (Polar) Form of a Complex Number
z = r(cos θ + i sin θ), where r = |z| and θ = arg(z)
Any complex number can be written using its modulus r and its argument (angle) θ, instead of its real and imaginary parts directly.
- r
- — the modulus of z, r = √(a²+b²)
- θ
- — the argument of z — the angle the line to z makes with the positive real axis
When to use it: Whenever a complex number needs to be expressed by its size and direction, especially before multiplying/dividing or raising to a power.
Worked Example
Convert a complex number to trigonometric form
Write z = 3+4i in trigonometric (polar) form.
Why Does This Work?
Since z=a+bi plots as the point (a,b), and r(cosθ) and r(sinθ) are exactly the horizontal and vertical coordinates of a point at distance r and angle θ from the origin (the same general trig ratio definition used for angles of any size), a=r cosθ and b=r sinθ — substituting these into a+bi gives r cosθ + i(r sinθ) = r(cosθ + i sinθ) directly.
Real-Life Example
Representing a rotating phasor in electrical engineering
An AC signal's magnitude and phase (timing offset) are both physically meaningful quantities.
Trigonometric form directly separates these — r is the signal's magnitude, θ is its phase — matching exactly how engineers think about AC signals as rotating "phasors."
Practice
Find the modulus r needed to write z = 6+8i in trigonometric form.
MediumCommon mistake
Using arctan alone without checking the actual quadrant of a+bi — arctan doesn't distinguish quadrants by itself, so θ must be adjusted to match where a+bi actually sits (the same basic acute angle idea used for trigonometric ratios of any angle).
Quick Review
- z = r(cosθ + i sinθ), where r=|z| and θ=arg(z).
- a = r cosθ and b = r sinθ.
- Always check the quadrant of a+bi when finding θ from arctan(b/a).