De Moivre’s Theorem Revision Content

De Moivre’s theorem establishes a connection between complex numbers and trigonometry.
Named after French mathematician Abraham De Moivre, it provides a method of raising complex numbers to a power.
The theorem states that for any real number n and any complex number z, where z = r(cosθ + isinθ), z^n = r^n(cos(nθ) + isin(nθ)).

Raise a complex number to a power: Convert the complex number into trigonometric form (r(cosθ + isinθ)), then apply De Moivre’s theorem.
Find roots of a complex number: Apply De Moivre’s theorem with the power being 1/n (the root you want to find). This involves finding n equally spaced points on the unit circle in the Argand plane.
Note that De Moivre’s theorem returns only one of the n roots, the others can be found by adding 360/n (or 2π/n in radians) to the argument and reducing to the range (-pi, pi].

De Moivre’s theorem can be used to derive many trigonometric identities, offering a new way to tackle trigonometric problems.
Cosine event formulae and sine event formulae are examples of identities that can be derived easily using De Moivre’s theorem.
To derive them, consider the multiple angles formulae for cosine and sine, which are derived directly as a result of De Moivre’s theorem.

Addition and subtraction of complex numbers involve working with the real and imaginary parts separately, while the formula provided by De Moivre’s theorem is specifically useful for raising complex numbers to a power.
De Moivre’s theorem makes complex number multiplication and division significantly simpler, as you multiply or divide the magnitudes and add or subtract the arguments respectively.
Working with power and roots of complex numbers becomes much easier under this theorem.