r(cosθ+isinθ) Calculator

Calculator Input

Modulus r

Angle θ

Angle Unit Degrees Radians

Decimal Precision

Power n for zⁿ

Root Count n

Calculate

Reset

Example Data Table

Formula Used

The trigonometric form of a complex number is z = r(cosθ + i sinθ).

Rectangular conversion uses a = r cosθ and b = r sinθ.

So the rectangular form becomes z = a + bi.

The conjugate is a - bi.

For powers, De Moivre’s theorem gives zⁿ = rⁿ(cos(nθ) + i sin(nθ)).

For nth roots, each root is r^(1/n)[cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)], where k = 0, 1, 2, ..., n-1.

How to Use This Calculator

1. Enter the modulus r.
2. Enter the angle θ.
3. Choose degrees or radians.
4. Set the decimal precision you want.
5. Enter n for power calculations.
6. Enter n for root calculations.
7. Click Calculate.
8. Read the rectangular form, conjugate, normalized angle, power result, and nth roots.
9. Use the CSV or PDF button to save the output.

About the r(cosθ+isinθ) Calculator

Why this form matters

The r(cosθ+isinθ) form is a standard way to write complex numbers. It is also called trigonometric form or cis form. This notation is useful in algebra, geometry, signals, and engineering. It separates size from direction. The modulus r shows distance from the origin. The angle θ shows rotation from the positive real axis.

What the calculator gives you

This calculator converts trigonometric form into rectangular form. It finds the real part and the imaginary part. It also shows the conjugate. You can view the angle in degrees and radians. The tool normalizes the angle too. That helps when your input is larger than one full turn or when it is negative.

Useful for powers and roots

Complex multiplication becomes easier in trigonometric form. Powers also become easier. De Moivre’s theorem uses the modulus and angle directly. This calculator applies that rule for zⁿ. It also lists nth roots. That saves time in exam work and checking. You can compare each root in trig form and rectangular form.

Helpful for learning and checking

Students often switch between a + bi and r(cosθ + i sinθ). This tool makes that process clear. It supports classroom examples, homework checking, and revision. The example data table gives quick practice values. The export buttons help you save work for notes, reports, or later review. The clean layout also keeps the input steps simple.

When to use it

Use this calculator when a problem gives modulus and argument. Use it when you need rectangular coordinates. Use it when solving power questions, root questions, and argument questions. It is also useful in vectors, phasors, and polar style complex number analysis. The output is direct, readable, and practical for daily study.

FAQs

1. What does r represent in this calculator?

r is the modulus of the complex number. It shows the distance from the origin to the point on the complex plane.

2. What does θ represent?

θ is the argument or angle of the complex number. It measures the direction from the positive real axis to the point.

3. Can I enter angles in radians?

Yes. Choose radians from the angle unit menu. The calculator will still show helpful degree and radian outputs after conversion.

4. What is the rectangular form output?

The rectangular form is a + bi. The calculator finds a = rcosθ and b = rsinθ, then displays the result clearly.

5. Why is the angle normalized?

Normalization places the angle in a standard range. That makes results easier to compare, interpret, and reuse in later steps.

6. What is the conjugate used for?

The conjugate helps with division, modulus checks, and simplification. If z = a + bi, then its conjugate is a - bi.

7. Does this calculator find powers and roots?

Yes. It calculates zⁿ with De Moivre’s theorem and lists nth roots using evenly spaced arguments around the complex plane.

8. When should I use CSV or PDF export?

Use CSV when you want table data for spreadsheets. Use PDF when you want a clean shareable copy for notes, classwork, or review.