Enter the magnitude of the complex number and the positive integer exponent into the calculator to calculate De Moivre’s Theorem.
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De Moivre’s Theorem Formula
The following formula is used to calculate De Moivre’s Theorem:
z^n = (r^n * (cos(nθ) + i * sin(nθ)))
Variables:
- z is a complex number
- n is a positive integer exponent
- r is the magnitude (absolute value) of the complex number z
- θ is the argument (angle) of the complex number z
To calculate De Moivre’s Theorem, raise the magnitude of the complex number to the power of the exponent. Multiply the result by the cosine of the product of the exponent and the argument, and add the product of the exponent and the sine of the argument multiplied by the imaginary unit (i).
What is De Moivre’s Theorem?
De Moivre’s Theorem is a formula in the field of complex numbers that connects trigonometry and complex numbers. Named after Abraham de Moivre, a French mathematician, this theorem is particularly useful in simplifying and solving equations involving complex numbers. The theorem states that for any real number x and any integer n, the formula (cos x + i sin x)^n = cos nx + i sin nx holds true, where i is the imaginary unit. This theorem allows for the easy calculation of powers and roots of complex numbers and is also used in deriving trigonometric identities. It is a key tool in the study of trigonometry, calculus, and complex number theory.
How to Calculate De Moivre’s Theorem?
The following steps outline how to use De Moivre’s Theorem to calculate complex number powers.
- First, write the complex number in polar form.
- Next, express the power as an exponent.
- Apply De Moivre’s Theorem: raise the modulus to the power and multiply the argument by the power.
- Convert the result back to rectangular form if necessary.
- Finally, simplify the expression if possible.
Example Problem:
Use the following variables as an example problem to test your knowledge.
Complex number in polar form: (z = 3(cos(frac{pi}{4}) + isin(frac{pi}{4})))
Power: (n = 5)
