Result
Enter values and press Calculate Complex Sine. Your result will appear here above the form.
Calculator form
Example data table
| Input form | Complex input z | sin(z) | Magnitude of sin(z) | Phase of sin(z) |
|---|---|---|---|---|
| Rectangular | 1 + 0i | 0.841471 + 0.000000i | 0.841471 | 0.000000 rad |
| Rectangular | 1 + 1i | 1.298458 + 0.634964i | 1.445397 | 0.454820 rad |
| Rectangular | 0 + 2i | 0.000000 + 3.626860i | 3.626860 | 1.570796 rad |
| Polar | 2∠45° | 2.151536 + 0.301761i | 2.172594 | 0.139345 rad |
Formula used
For a complex number z = a + bi, the complex sine is:
sin(a + bi) = sin(a)cosh(b) + i·cos(a)sinh(b)
Real output = sin(a)cosh(b), imaginary output = cos(a)sinh(b).
When you enter polar data, the calculator first converts it into rectangular form:
a = r cos(θ)
b = r sin(θ)
It then applies the complex sine identity and also reports output magnitude and phase.
How to use this calculator
- Choose Rectangular form for real and imaginary parts.
- Choose Polar form for magnitude and angle.
- Select the correct angle unit for polar input.
- Pick the decimal precision you want in the result.
- Press Calculate Complex Sine.
- Read the result, detailed steps, and complex plane graph.
- Use the CSV or PDF buttons to save the output.
FAQs
1) What does the calculator compute?
It computes the sine of a complex number z. The output includes real part, imaginary part, magnitude, phase, and a graph on the complex plane.
2) Can I enter values in polar form?
Yes. Choose polar mode, then enter magnitude and angle. The tool converts that input into rectangular form before applying the complex sine formula.
3) Which formula is used for complex sine?
The calculator uses sin(a + bi) = sin(a)cosh(b) + i·cos(a)sinh(b). This identity separates the real and imaginary components clearly and accurately.
4) Why are hyperbolic functions included?
Complex trigonometric identities naturally involve hyperbolic functions when the imaginary component is present. That is why cosh and sinh appear in the final expression.
5) What is the phase of the result?
The phase is the output angle of sin(z) on the complex plane. It is computed with atan2 of the imaginary and real result parts.
6) Does the graph show both input and output?
Yes. The graph plots the original complex number and the computed sine value. This helps you compare direction and size visually.
7) Can I download the result?
Yes. Use the CSV button for spreadsheet-friendly data and the PDF button for a compact report you can save or share.
8) When should I use rectangular instead of polar input?
Use rectangular form when you already know a and b. Use polar form when magnitude and angle are easier or more natural from the problem statement.