De Moivre´s theorem

De Moivre's Theorem in Math

Definition

De Moivre's theorem is a mathematical theorem that relates complex numbers to trigonometry. It provides a way to raise a complex number to a power, which is particularly useful when dealing with complex exponentiation.

Knowledge Points

De Moivre's theorem contains the following key points:

1. Complex numbers
2. Trigonometry
3. Exponentiation of complex numbers

Formula

The formula for De Moivre's theorem is as follows:

where:

• is the angle in radians
• is the power to which the complex number is raised

Application

To apply De Moivre's theorem, follow these steps:

1. Express the complex number in trigonometric form: , where is the magnitude of the complex number.
2. Raise the trigonometric form to the desired power using De Moivre's theorem formula.
3. Simplify the resulting expression by expanding and combining like terms.
4. Convert the expression back to rectangular form if needed.

Symbol

There is no specific symbol for De Moivre's theorem. It is commonly referred to as "De Moivre's theorem" or "De Moivre's formula."

Methods

There are several methods to apply De Moivre's theorem, including:

1. Directly applying the formula and simplifying the expression.
2. Using the polar form of complex numbers to simplify calculations.
3. Utilizing the periodicity of trigonometric functions to find patterns in the powers.

Solved Examples

1. Example 1: Find if ![z = 2(cos(pi/3) + isin(pi/3))]. Solution:

   • Convert to rectangular form: ![z = 2(cos(pi/3) + isin(pi/3)) = 2(1/2 + i(sqrt(3)/2)) = 1 + i(sqrt(3))].
   • Apply De Moivre's theorem: ![z^4 = (1 + i(sqrt(3)))^4].
   • Expand and simplify: ![z^4 = 1 + 4i(sqrt(3)) - 6 + 4i(sqrt(3)) = -5 + 8i(sqrt(3))].

2. Example 2: Find if ![z = 3(cos(pi/4) + isin(pi/4))]. Solution:

   • Convert to rectangular form: ![z = 3(cos(pi/4) + isin(pi/4)) = 3(1/sqrt(2) + i(1/sqrt(2))) = 3/sqrt(2) + 3i/sqrt(2)].
   • Apply De Moivre's theorem: ![z^6 = (3/sqrt(2) + 3i/sqrt(2))^6].
   • Expand and simplify: ![z^6 = 729/8 + 729i/8].

Practice Problems

1. Find if ![z = 4(cos(pi/6) + isin(pi/6))].
2. Find if ![z = 5(cos(pi/2) + isin(pi/2))].
3. Find if ![z = 2(cos(pi/8) + isin(pi/8))].

FAQ

Q: What is De Moivre's theorem? De Moivre's theorem is a mathematical theorem that relates complex numbers to trigonometry. It provides a way to raise a complex number to a power.

Q: How do I apply De Moivre's theorem? To apply De Moivre's theorem, convert the complex number to trigonometric form, raise it to the desired power using the formula, simplify the expression, and convert it back to rectangular form if needed.

Q: Can De Moivre's theorem be used for negative powers? Yes, De Moivre's theorem can be used for negative powers as well. Simply apply the formula and simplify the expression accordingly.