Complex Number To Trigonometric Form Calculator

Converting Complex Numbers to Trigonometric Form

Every complex number z = a + bi can be expressed in trigonometric (polar) form as z = r(cosθ + i sinθ), where r is the modulus and θ is the argument. This representation separates a complex number's magnitude from its direction, making multiplication, division, exponentiation, and root extraction far more efficient than working in rectangular form.

The Core Formula

The conversion from rectangular to trigonometric form rests on two calculations:

• Modulus (r): r = √(a² + b²) — the Euclidean distance from the origin to the point (a, b) in the complex plane.
• Argument (θ): θ = atan2(b, a) — the angle measured counterclockwise from the positive real axis, correctly resolved across all four quadrants.

The complete trigonometric form is written z = r(cosθ + i sinθ), sometimes abbreviated as r cis θ.

Variable Definitions

Real part (a): The horizontal coordinate in the complex plane. For z = 3 + 4i, a = 3.

Imaginary part (b): The vertical coordinate. For z = 3 + 4i, b = 4.

Modulus (r): Always non-negative. For z = 3 + 4i: r = √(9 + 16) = √25 = 5.

Argument (θ): Expressed in the range (−π, π] radians or (−180°, 180°]. For z = 3 + 4i: θ = atan2(4, 3) ≈ 0.9273 rad ≈ 53.13°.

Geometric Derivation

Plotting z = a + bi as a point in the complex plane creates a right triangle with legs of length |a| and |b| and hypotenuse r. The angle θ satisfies cosθ = a/r and sinθ = b/r. Substituting a = r cosθ and b = r sinθ back into a + bi yields z = r cosθ + i(r sinθ) = r(cosθ + i sinθ).

The atan2(b, a) function is essential because the simpler arctan(b/a) only returns values in (−90°, 90°) and cannot distinguish quadrant I from quadrant III. For example, z = 1 + i and z = −1 − i both produce arctan(1) = 45°, but their correct arguments are 45° and −135° respectively. According to Paul Dawkins, Polar and Exponential Forms (Lamar University), correctly identifying the quadrant is essential for an unambiguous argument value.

Worked Examples

Example 1 — First quadrant: z = 1 + i. r = √2 ≈ 1.414; θ = atan2(1, 1) = 45°. Result: z = √2(cos 45° + i sin 45°).

Example 2 — Second quadrant: z = −3 + 4i. r = √(9 + 16) = 5; θ = atan2(4, −3) ≈ 126.87°. Result: z = 5(cos 126.87° + i sin 126.87°).

Example 3 — Third quadrant: z = −1 − i. r = √2; θ = atan2(−1, −1) = −135°. Result: z = √2(cos(−135°) + i sin(−135°)).

Example 4 — Purely imaginary: z = 5i. r = 5; θ = atan2(5, 0) = 90°. Result: z = 5(cos 90° + i sin 90°).

Practical Applications

• AC circuit analysis: Impedance Z = R + jX converts to polar form Z = |Z|∠φ, directly revealing magnitude and phase angle for power factor calculations.
• De Moivre theorem: z^n = r^n(cos nθ + i sin nθ) reduces computing powers like z^10 to simple arithmetic on r and θ.
• nth roots: The k-th root is r^(1/n)(cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)) for k = 0, 1, ..., n−1, yielding exactly n distinct complex values.
• 2D rotation: Multiplying any complex number by cosα + i sinα rotates its corresponding point by α radians around the origin.

The Thiel College Mathematics Project (Trigonometry Using Complex Numbers) and Palomar College (Complex Numbers and Trigonometric Identities) both demonstrate that trigonometric form is indispensable for proving angle-addition identities and for applying Fourier analysis methods in signal processing.

Converting Back to Rectangular Form

To recover a + bi from r(cosθ + i sinθ), compute a = r cosθ and b = r sinθ. For z = 5(cos 53.13° + i sin 53.13°): a = 5 × 0.6 = 3 and b = 5 × 0.8 = 4, recovering z = 3 + 4i.