Convert a complex number from rectangular form a + bi to trigonometric (polar) form r(cos θ + i sin θ). Shows modulus, argument, exponential form, quadrant, reciprocal, and De Moivre powers.
The **Complex Number to Trigonometric Form Calculator** converts any complex number written in rectangular form a + bi into its trigonometric (polar) representation r·(cos θ + i·sin θ). Enter the real part a and imaginary part b, and the tool computes the modulus r = √(a² + b²), the argument θ = atan2(b, a), and displays the result in trigonometric, exponential (Euler), and rectangular form all at once.
Converting from rectangular to polar form is a fundamental skill in complex analysis and electrical engineering. Phasor arithmetic in AC circuits requires polar form for efficient multiplication and division. Signal processing uses the modulus as amplitude and the argument as phase. In pure mathematics, De Moivre's theorem — zⁿ = rⁿ(cos nθ + i sin nθ) — makes exponentiation trivial once you have polar coordinates.
The calculator identifies the quadrant, computes the reciprocal 1/z, and provides |z|² for quick magnitude comparisons. An interactive De Moivre section shows the 2nd through 6th powers of your number, revealing how repeated multiplication rotates and scales the point in the complex plane. Visual bars compare the real part, imaginary part, and modulus side by side.
A reference table of ten standard complex ↔ polar conversions — from axis points like i and −1 to classic Pythagorean triples like 3 + 4i = 5∠53.13° — provides benchmarks you can verify in one click with the presets.
Complex Number to Trigonometric Form Calculator (a + bi → Polar) helps you avoid repetitive setup mistakes when solving trigonometric and coordinate-geometry problems. Instead of recalculating conversions, signs, and edge cases by hand, you can test inputs immediately, inspect intermediate values, and confirm final answers before submitting work or using numbers in downstream calculations. It surfaces key outputs like Trigonometric Form, Modulus (r), Argument (degrees) in one pass.
Given z = a + bi: Modulus r = √(a² + b²). Argument θ = atan2(b, a). Trig form: z = r(cos θ + i sin θ). Exponential form: z = r·e^(iθ). Powers: zⁿ = rⁿ(cos nθ + i sin nθ).
Result: Computed from the entered values
Using a=1, b=0, the calculator returns Computed from the entered values. This example mirrors the calculator's live computation flow and is useful for checking manual steps and unit handling.
This calculator is tailored to complex number to trigonometric form calculator (a + bi → polar) workflows, including common input modes, unit handling, and special-case behavior. It is designed for fast checking during homework, exam preparation, technical drafting, and coding tasks where trigonometric consistency matters.
Use the primary result together with supporting outputs to verify direction, magnitude, and validity. Cross-check against known identities or geometric constraints, and confirm that angle ranges, sign conventions, and domain restrictions are satisfied before using the numbers elsewhere.
A reliable way to improve is to solve once manually, then verify with the calculator and explain any mismatch. Repeat this on varied examples and edge cases. The built-in preset scenarios for quick trials, comparison tables for side-by-side validation, visual cues that make trends and quadrants easier to read help you build pattern recognition and reduce sign or conversion errors over time.
It is z = r(cos θ + i sin θ), also written r·cis θ, where r is the modulus (distance from origin) and θ is the argument (angle from the positive real axis). Use this as a practical reminder before finalizing the result.
Compute r = √(a² + b²), where a is the real part and b is the imaginary part of the complex number. Keep this note short and outcome-focused for reuse.
Use θ = atan2(b, a). This gives the correct angle in all four quadrants, unlike atan(b/a) which only covers (−90°, 90°).
It states that zⁿ = rⁿ(cos nθ + i sin nθ). This makes computing powers and roots of complex numbers straightforward when in polar form.
They are equivalent: r(cos θ + i sin θ) = r·e^(iθ) by Euler's formula. The exponential form is more compact and convenient for calculus operations.
Only the number 0 + 0i has modulus zero. Its argument is undefined. All other complex numbers have a positive modulus and a unique argument (up to multiples of 360°).