The formula to calculate De Moivre's Theorem is:
\[ z^n = r^n \left( \cos(n\theta) + i \sin(n\theta) \right) \]
Where:
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.
Let's assume the following values:
Using the formula to calculate De Moivre's Theorem:
\[ z^n = 2^3 \left( \cos(3 \cdot \frac{\pi}{4}) + i \sin(3 \cdot \frac{\pi}{4}) \right) \]
Calculating the components:
\[ r^n = 2^3 = 8 \]
\[ \cos(3 \cdot \frac{\pi}{4}) = \cos(\frac{3\pi}{4}) = -\frac{\sqrt{2}}{2} \]
\[ \sin(3 \cdot \frac{\pi}{4}) = \sin(\frac{3\pi}{4}) = \frac{\sqrt{2}}{2} \]
So the result is:
\[ z^n = 8 \left( -\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2} \right) = -4\sqrt{2} + 4\sqrt{2}i \]
The result of De Moivre's Theorem is \( -4\sqrt{2} + 4\sqrt{2}i \).
