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Complex Number to Trigonometric Form Calculator.

Write a complex number as r(cos θ + i sin θ).

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Modulus: 5

Modulus

0.000000
Argument: 0.927295 rad

Argument

0.000000rad
Argument: 53.130102 °

Argument

0.000000°
Trigonometric form: 5 (cos(0.9272952180016122) + i·sin(0.9272952180016122))

Trigonometric form

5 (cos(0.9272952180016122) + i·sin(0.9272952180016122))

Results update automatically as you type.

Use Cases

Simplify complex number operations

Convert to trigonometric form to easily multiply, divide, or raise complex numbers to powers using De Moivre's theorem.

Example: Multiply (1 + i) and (√3 - i) by converting each to polar form first.

Visualize complex numbers on the plane

Understand the geometric representation of a complex number as a point with distance r from origin and angle θ from the real axis.

Example: Plot 3 + 4i as r=5 and θ≈53.13°.

Frequently Asked Questions

What is the trigonometric form of a complex number?
The trigonometric form (or polar form) expresses a complex number as r(cos θ + i sin θ), where r is the modulus (distance from origin) and θ is the argument (angle with the positive real axis).
How do I find the modulus and argument?
Given real part a and imaginary part b, the modulus r = √(a² + b²). The argument θ is found using tan⁻¹(b/a), adjusted for the quadrant of the point (a, b).
Can the argument be negative?
Yes, the argument is often given in the range (-π, π] or [0, 2π). This calculator provides the principal value, which may be negative if the complex number lies below the real axis.

Tips & Common Mistakes

Tips

  • Ensure the real and imaginary parts are entered as decimal numbers or integers; the calculator handles both.
  • Remember that the argument is measured in radians by default; you can convert to degrees if needed.
  • For negative real parts, the argument is adjusted by adding π to the arctangent result to place it in the correct quadrant.
  • Use the trigonometric form to quickly compute powers: (r(cos θ + i sin θ))ⁿ = rⁿ(cos(nθ) + i sin(nθ)).

Common Mistakes to Avoid

  • Forgetting to adjust the argument for the correct quadrant when the real part is negative, leading to an angle that points in the wrong direction.
  • Using degrees instead of radians without conversion, which yields incorrect trigonometric values.
  • Mixing up the real and imaginary parts when calculating the modulus; r is always √(a² + b²), not √(a² - b²).

Last updated: August 13, 2026