De Moivre’s Theorem Calculator

Need to calculate powers of complex numbers? The De Moivre’s Theorem Calculator is here to help. Enter the complex number, exponent, magnitude and argument required values into the calculator to find the complex number raised to a given power.

De Moivre’s Theorem Calculator

Enter any 3 values (Complex number, exponent, magnitude, or argument) to calculate the missing one. Advanced mode allows additional options for calculation.

Enter the complex number (z) to calculate its power using De Moivre’s theorem. Enter the exponent (n) to which you want to raise the complex number. Enter the magnitude (r) of the complex number. Enter the argument (θ) in radians for the complex number. The result will be displayed here after calculation.

What is De Moivre’s Theorem?

De Moivre’s Theorem is a formula that links complex numbers and trigonometry. It allows you to raise a complex number to any power using polar coordinates.

Guide For User

To use the De Moivre’s Theorem Calculator is straightforward. Here’s how:

Complex Number (z): Enter the complex number. For example, type 2 + 3i.
Exponent (n): Enter the exponent. For instance, type 4.
Magnitude (r): Enter the magnitude. Try 3.6.
Argument (θ): Enter the argument in radians. Enter 0.927.

Click “Calculate” to get the result.

Formula

The formula for De Moivre’s Theorem is:

z^{n} = (r^{n}) \cdot (\cos (n θ) + i \sin (n θ))

Variable	Description
$z$	Complex number
$n$	Exponent
$r$	Magnitude of the complex number
$θ$	Argument (angle) in radians

How To calculate – Examples

Example 1

More Calculators

Complex Number: $2 + 3 i$
Exponent: $4$
Magnitude: $3.6$
Argument: $0.927$

Step-by-Step Calculation:

1. Identify the values: $z = 2 + 3$ , $n = 4$ , $r = 3.6$ , $θ = 0.927$

2. Calculate the new magnitude:

r^{n} = 3. 6^{4} = 167.9616

3. Calculate the new argument:

n θ = 4 \times 0.927 = 3.708

4. Find the real part:

167.9616 \cdot \cos (3.708) = - 147.310

5. Find the imaginary part:

167.9616 \cdot \sin (3.708) = - 83.983

Result: $- 147.310 + - 83.983 i$

Example 2

Complex Number: $1 + i$
Exponent: $3$
Magnitude: $1.414$
Argument: $0.785$

Step-by-Step Calculation:

1. Identify the values: $z = 1 + i$ , $n = 3$ , $r = 1.414$ , $θ = 0.785$

2. Calculate the new magnitude:

r^{n} = 1.41 4^{3} = 2.828

3. Calculate the new argument:

n θ = 3 \times 0.785 = 2.355

3. Find the real part:

2.828 \cdot \cos (2.355) = - 2.0

4. Find the imaginary part:

2.828 \cdot \sin (2.355) = 2.828

Result: $- 2.0 + 2.828 i$

I hope by using our De Moivre’s Theorem Calculator, you did effortlessly raised any complex number to a given power. Please lets us know on report error page if you are facing any issue while using the tool.

De Moivre’s Theorem Calculator