Complex Number Cube Root Calculation
To find the cube roots of a complex number, first, convert the complex number into trigonometric form. Then calculate the magnitude and phase angle of the complex number. The cube roots will have different phase angles due to the periodic nature of trigonometric functions.
De Moivre's Formula
The De Moivre's Formula helps in finding the roots of complex numbers by utilizing the polar form of complex numbers. It states that for a complex number z = r(cos θ + i sin θ), the nth root of z is given by z^(1/n) = r^(1/n) [cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)], where k is an integer between 0 and n-1.
Example Calculation:Let's consider a complex number z = 27(cos 327° + i sin 327°). We need to find the cube roots of this complex number using De Moivre's Formula.
The magnitude of z is 27 and the phase angle θ is 2.268 radians (approximately 1.817π).
By substituting the values in the formula for k = 0, 1, 2, we find the cube roots as follow:
- For k = 0: z^(1/3) = -0.978 + i 2.836
- For k = 1: z^(1/3) = -1.967 - i 2.265
- For k = 2: z^(1/3) = 2.945 - i 0.571
Therefore, the cube roots of the complex number 27(cos 327° + i sin 327°) are -0.978 + i 2.836, -1.967 - i 2.265, and 2.945 - i 0.571 respectively.