Calculator inputs
Example presets
Formula used
Real part: a = r cos(θ)
Imaginary part: b = r sin(θ)
Cartesian form: z = a + bi
The calculator first converts the selected angle unit into radians. It then evaluates cosine and sine before multiplying both values by the magnitude.
How to use this calculator
- Choose polar, trigonometric, exponential, or phasor notation.
- Select separate fields or full-expression parsing.
- Enter the magnitude and angle, including π expressions when needed.
- Choose degrees, radians, gradians, or turns.
- Set precision, rounding, output notation, and display preferences.
- Select Convert to Cartesian form to view the result, steps, graph, conjugate, and equivalent forms.
- Use the copy, CSV, PDF, or print controls to save the calculation.
Worked examples
| Input | Real calculation | Imaginary calculation | Cartesian form |
|---|---|---|---|
| 5∠30° | 5 cos 30° = 5√3/2 | 5 sin 30° = 5/2 | 5√3/2 + 5/2i |
| 4∠90° | 4 cos 90° = 0 | 4 sin 90° = 4 | 4i |
| 2∠−45° | 2 cos −45° = √2 | 2 sin −45° = −√2 | √2 − √2i |
| 3e^(iπ/3) | 3 cos π/3 = 3/2 | 3 sin π/3 = 3√3/2 | 3/2 + 3√3/2i |
Common-angle reference
| Angle | Radians | cos θ | sin θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | π/6 | √3/2 | 1/2 |
| 45° | π/4 | √2/2 | √2/2 |
| 60° | π/3 | 1/2 | √3/2 |
| 90° | π/2 | 0 | 1 |
| 180° | π | −1 | 0 |
| 270° | 3π/2 | 0 | −1 |
| 360° | 2π | 1 | 0 |
Understanding complex Cartesian conversion
A complex number combines a real coordinate with an imaginary coordinate. Cartesian form writes those coordinates as a plus bi, so the number can be placed directly on the complex plane. This form makes addition, subtraction, comparison, and graphing especially straightforward.
Polar form describes the same number using a magnitude and angle. The magnitude measures distance from the origin, while the angle shows the vector direction from the positive real axis. Converting between these forms connects geometry with algebraic complex-number operations.
The real coordinate comes from multiplying the magnitude by cosine. The imaginary coordinate comes from multiplying the magnitude by sine. These two projections resolve one angled vector into perpendicular horizontal and vertical components.
Angles can be entered in degrees, radians, gradians, or turns. Degrees are common in elementary problems, while radians appear frequently in calculus, engineering, physics, and signal analysis. The calculator normalizes every unit before applying trigonometric functions accurately.
Negative angles rotate clockwise, and positive angles rotate counterclockwise. Angles beyond one revolution remain valid because coterminal angles represent the same final direction. Normalized and principal arguments help compare equivalent representations without changing values.
Small floating-point remnants may appear near axes after computation. The near-zero threshold converts those insignificant values into zero, preventing results such as 6.123234e-17i when the intended value is exactly zero. Precision and rounding controls then shape the displayed answer for practical use.
Exact output is especially useful for standard angles with known radicals. For example, cosine thirty degrees equals square root three over two, while sine thirty degrees equals one half. Exact forms preserve mathematical structure before decimal approximations are introduced.
The Argand diagram provides a visual check of the conversion. A point at a plus bi appears at horizontal coordinate a and vertical coordinate b, while the connecting vector shows magnitude and direction. This view helps identify quadrants, axes, conjugates, and rotational symmetry.
Cartesian results are widely used in electrical engineering, control systems, quantum mechanics, Fourier analysis, and communications. Phasors representing alternating signals often begin in magnitude-angle notation and are converted before component calculations. Reliable conversion supports clearer calculations across many technical applications.
Frequently asked questions
1. What is Cartesian form?
Cartesian form writes a complex number as a + bi, where a is the real part and b is the imaginary part.
2. Can I enter π in the angle field?
Yes. Expressions such as π/4, 3π/2, and −π/6 are accepted when radians are selected.
3. Are negative angles supported?
Yes. Negative angles rotate clockwise and are converted normally.
4. Can angles exceed one revolution?
Yes. The calculator accepts large positive or negative angles and also displays a normalized angle.
5. Why does a tiny value become zero?
The near-zero threshold removes harmless floating-point remnants that should represent exact zero.
6. What is the principal argument?
It is the representative angle restricted to a standard interval, shown here from −180° through 180°.
7. What is the conjugate?
The conjugate keeps the real part and reverses the sign of the imaginary part.
8. Can the result use scientific notation?
Yes. Select scientific or engineering notation from the number-format menu.
9. Can I export the calculation?
Yes. The result can be copied, downloaded as CSV, generated as a PDF, or printed.