Complex Number to Sine Cosine Calculator

Calculator

Real Part (a)

Imaginary Part (b)

Decimal Precision

Display Angle In

Normalize angle to 0 through 360 degrees

Plotly Graph

The graph shows the vector from the origin to the complex number. It also traces the angle sweep used in trigonometric form.

Example Data Table

Complex Number	Modulus	Angle (Degrees)	Angle (Radians)	Sine Cosine Form
3 + 4i	5	53.1301	0.9273	5 [cos(53.1301°) + i sin(53.1301°)]
1 + 1i	1.4142	45	0.7854	1.4142 [cos(45°) + i sin(45°)]
-2 + 2i	2.8284	135	2.3562	2.8284 [cos(135°) + i sin(135°)]
4 - 4i	5.6569	315	5.4978	5.6569 [cos(315°) + i sin(315°)]

Formula Used

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

r = √(a² + b²)

The argument or angle is:

θ = atan2(b, a)

The sine cosine form becomes:

z = r [cos(θ) + i sin(θ)]

Here, cos(θ) = a / r and sin(θ) = b / r when r ≠ 0.

How to Use This Calculator

Enter the real component in the first field.
Enter the imaginary component in the second field.
Choose how many decimal places you want.
Select degrees or radians for angle display.
Keep normalization checked for principal positive angles.
Press Convert Now to generate the result.
Review the trig form, modulus, angle, and graph.
Download the generated row as CSV or PDF.

Frequently Asked Questions

1. What does sine cosine form mean?

Sine cosine form writes a complex number using modulus and angle. It shows the number as a distance from the origin and a rotation from the positive real axis.

2. Why is atan2 used for the angle?

atan2(b, a) finds the correct angle using both signs. It places the angle in the correct quadrant, unlike a basic inverse tangent formula.

3. What is the modulus of a complex number?

The modulus is the distance from the origin to the point (a, b). It is computed with the square root of a² + b².

4. Can the angle be shown in radians?

Yes. The calculator supports degrees and radians. Choose your preferred display unit before converting the complex number.

5. What happens when the number is zero?

For 0 + 0i, the modulus is zero. The angle is not unique, so the calculator displays a default trigonometric representation using zero magnitude.

6. Why normalize the angle?

Normalization converts negative angles into equivalent positive ones between zero and 360 degrees. This often makes comparison and reporting easier.

7. Is this the same as polar form?

Yes. Sine cosine form is a trigonometric version of polar form. Both use magnitude and argument to represent the same complex number.

8. When is trig form useful?

Trig form helps with multiplication, division, powers, and roots of complex numbers. It is especially helpful in mathematics, engineering, and signal analysis.