De Moivre's Theorem

De Moivre’s Theorem tells you how to raise a complex number on the unit circle to a whole-number power by multiplying its angle instead of expanding the binomial. It works because multiplication of complex numbers in trigonometric form adds arguments and multiplies magnitudes, so repeated multiplication by the same number keeps adding the same angle each time. This makes powers of complex numbers much easier to compute, and it also links trigonometry with algebraic identities for sine and cosine.

Use this when a complex number is written in polar/trigonometric form and you need to find a large power or simplify expressions involving (cosθ + i sinθ). It is especially useful in complex numbers, trigonometric proofs, and deriving multiple-angle formulas.

• Multiplying the sine and cosine values separately instead of multiplying the angle by n
• Forgetting that the theorem applies directly to trig form, not an arbitrary complex number unless it is first converted
• Using the wrong angle units or sign, especially when the angle is in degrees versus radians