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 to calculate the missing variable

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:

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

zn=(rn)⋅(cos⁡(nθ)+isin⁡(nθ))
Variable Description
z

 

More Calculators
 Complex number
n

 

 Exponent
r

 

 Magnitude of the complex number
θ

 

 Argument (angle) in radians

How To calculate – Examples

Example 1

  • Complex Number:
    2+3i

     

  • 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:

          rn=3.64=167.9616

3. Calculate the new argument:

        nθ=4×0.927=3.708

4. Find the real part:

       167.9616⋅cos⁡(3.708)=−147.310

5. Find the imaginary part:

      167.9616⋅sin⁡(3.708)=−83.983

Result:

−147.310+−83.983i

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:

      rn=1.4143=2.828

3. Calculate the new argument:

      nθ=3×0.785=2.355

3. Find the real part:

       2.828⋅cos⁡(2.355)=−2.0

4. Find the imaginary part:

      2.828⋅sin⁡(2.355)=2.828

Result:

−2.0+2.828i

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.

 

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