Convert to Trigonometric Form Calculator

Calculator

Real part a

Imaginary part b

Angle unit

Angle range

Output style

Decimal precision

Example Data Table

Real a	Imaginary b	Modulus r	Argument	Quadrant or axis	Trigonometric form
3	4	5	53.1301 Degrees	Quadrant I	5[cos(53.1301°) + i sin(53.1301°)]
-5	12	13	112.6199 Degrees	Quadrant II	13[cos(112.6199°) + i sin(112.6199°)]
0	-7	7	270 Degrees	Negative imaginary axis	7 cis(270°)
1	-1.7321	2	300 Degrees	Quadrant IV	2[cos(300°) + i sin(300°)]

Formula Used

For a complex number z = a + bi, the trigonometric form is:

z = r[cos(θ) + i sin(θ)]

The modulus is:

r = √(a² + b²)

The argument is:

θ = atan2(b, a)

The atan2 function checks signs of both parts. It places the angle in the correct quadrant. This is safer than using tan⁻¹(b / a) alone.

How to Use This Calculator

Enter the real part of the complex number.
Enter the imaginary part without the letter i.
Choose degrees or radians for the angle.
Select the angle range required by your work.
Choose expanded form or cis shorthand.
Set decimal precision for rounding.
Press Calculate to see the result below the header.
Use CSV or PDF buttons to save the result.

Convert to Trigonometric Form Guide

Complex numbers become easier to compare when written in trigonometric form. The form links a rectangular value to distance and direction. A number a plus bi becomes r times the cosine of theta plus i sine theta. The modulus r measures distance from the origin. The argument theta measures rotation from the positive real axis.

Why This Conversion Matters

Trigonometric form is useful in algebra, engineering, signals, and geometry. Multiplication becomes cleaner because moduli multiply and arguments add. Division also becomes easier because moduli divide and arguments subtract. Powers are handled with De Moivre ideas. Roots are easier to list because angles can be spaced evenly around a circle.

Understanding the Inputs

The real part places the point left or right. The imaginary part places it up or down. Together they define one point on the complex plane. The calculator uses both values to find the modulus with the distance formula. It then uses atan2 to choose the correct quadrant. This avoids common sign mistakes.

Reading the Output

A positive modulus is shown first. The angle follows in degrees or radians. You can choose a standard angle range. The expanded expression shows cosine and sine terms. The cis style gives a shorter form. Both styles mean the same value. The table also shows quadrant, reference angle, and conjugate details.

Accuracy and Learning

Precision control helps match classroom or technical needs. Use more decimals for engineering work. Use fewer decimals for quick checking. Zero needs special care because its direction is not unique. For zero, the angle is undefined, and the trigonometric form is simply zero.

Practical Uses

This conversion supports phasors, rotations, wave analysis, and polar graphing. It also helps students see how rectangular coordinates connect with circular motion. The result can be saved as CSV for spreadsheets. The PDF option creates a compact report for notes, assignments, or review.

Checking Results

Review the sign of each input before trusting an angle. Points in different quadrants may share the same reference angle. They do not share the same argument. Compare the rectangular point with the reported quadrant. Then convert the result back mentally or with a calculator. This habit builds confidence and catches entry errors.

FAQs

What is trigonometric form?

Trigonometric form writes a complex number as r[cos(θ) + i sin(θ)]. It uses distance from the origin and angle from the positive real axis.

What do a and b mean?

The value a is the real part. The value b is the imaginary coefficient. Together, they describe the complex number a + bi.

How is the modulus found?

The modulus is found with r = √(a² + b²). It is always nonnegative and shows the point distance from the origin.

Why use atan2?

atan2 checks both signs before choosing the angle. It avoids quadrant errors that may happen when using tan⁻¹(b / a).

Can the angle be negative?

Yes. If you choose the -π to π range, some arguments can be negative. The value still represents the same complex direction.

What happens when the complex number is zero?

The modulus is zero. The argument is undefined because zero has no unique direction. The trigonometric form is simply zero.

What is cis notation?

cis θ is shorthand for cos(θ) + i sin(θ). So r cis θ means the same thing as r[cos(θ) + i sin(θ)].

Why export the result?

CSV export helps spreadsheet work. PDF export gives a compact report for assignments, records, or later review.