Complex Number to Trigonometric Form Calculator
Enter a complex number in rectangular form (a + bi) and get its trigonometric form r(cos θ + i·sin θ), also written r·cis(θ). The calculator shows the modulus r, argument θ, and the complete trig expression.
r = √(a² + b²) — the amplitude of the trigonometric form
- 1
Sum of squares
3² + 4² = 25 - 2
Argument θ
atan2(4, 3) = 53.1301°Used directly in the trig form as the phase angle. - 3
Modulus r
√25 = 5
How does this calculator work?
The trigonometric form of a + bi is r(cos θ + i·sin θ), where r = √(a² + b²) and θ = atan2(b, a). Enter the real and imaginary parts to get r, θ in degrees and radians, and the complete trigonometric expression.
Formula
How this is calculated
The trigonometric (or polar-trigonometric) form of a complex number writes it explicitly in terms of cosine and sine rather than the compact r∠θ notation. Starting from a + bi, the modulus r = √(a² + b²) gives the amplitude and the argument θ = atan2(b, a) gives the angle. Substituting back yields z = r(cos θ + i·sin θ), which can be verified by expanding: r·cos θ = a and r·sin θ = b by the definition of polar coordinates.
This form is equivalent to the polar form r∠θ and the exponential form r·e^(iθ) via Euler's formula (e^(iθ) = cos θ + i·sin θ). The compact notation r·cis(θ) is a shorthand for r(cos θ + i·sin θ) used in some textbooks.
The trigonometric form makes De Moivre's theorem particularly transparent: [r(cos θ + i·sin θ)]^n = r^n(cos nθ + i·sin nθ), which is the foundation for finding complex powers and roots.
Frequently asked questions
They represent the same thing expressed differently. Polar form r∠θ is a compact notation, while trigonometric form r(cos θ + i·sin θ) writes out the cosine and sine components explicitly. Both contain the modulus and argument.
cis(θ) is a shorthand notation for cos(θ) + i·sin(θ). It is used in many textbooks to write the trigonometric form more compactly as r·cis(θ).
To multiply two complex numbers in trigonometric form, multiply their moduli and add their arguments: r₁·cis(θ₁) × r₂·cis(θ₂) = r₁r₂·cis(θ₁ + θ₂). This is far simpler than expanding (a₁ + b₁i)(a₂ + b₂i) in rectangular form.
TG we-Calculate Editorial Team. (2026). Complex Number to Trigonometric Form Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/complex-number-to-trigonometric-form-calculator
TG we-Calculate Editorial Team. "Complex Number to Trigonometric Form Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/complex-number-to-trigonometric-form-calculator.
TG we-Calculate Editorial Team, "Complex Number to Trigonometric Form Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/complex-number-to-trigonometric-form-calculator
@misc{wecalculate_complex_number_to_trigonometric_form_calculator, title = {Complex Number to Trigonometric Form Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/complex-number-to-trigonometric-form-calculator}}, year = {2026}, note = {TG we-Calculate} }
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