De Moivre's formula

De Moivre's formula states that \((\cos \theta+i\sin \theta)^n=\cos(n\theta)+i\sin(n\theta)\) for integer \(n\), linking powers of complex numbers to multiplication of angles.

Complex number: A number of the form \(a+bi\), where \(a,b\) are real and \(i^2=-1\). The formula applies to complex numbers expressed through their magnitude and angle.
Imaginary unit: The number \(i\) satisfying \(i^2=-1\). The formula's trigonometric form uses \(i\) to encode the imaginary component.
Roots of unity: Complex numbers whose positive integer powers equal one. Setting the powered expression equal to one gives their angles directly.
Abraham de Moivre: A French-born mathematician whose work included probability theory and a formula for powers of trigonometric expressions. The identity bears his name and appeared in his mathematical work.
Polar form: A representation of a complex number by its magnitude and angle from the positive real axis. Writing a complex number in polar form exposes the angle multiplication in the identity.
Sine and cosine: The basic trigonometric functions that assign coordinates on the unit circle to an angle. They provide the real and imaginary coordinates appearing on both sides of the identity.
Complex nth root: A complex number whose nth power equals a specified complex number. De Moivre's formula converts root extraction into dividing angles and accounting for multiple branches.
Mathematical notation: A system of symbols and conventions used to express mathematical ideas. The compact modern statement reflects notation developed after the formula's early use.
Complex multiplication: Multiplication of complex numbers, which combines their magnitudes multiplicatively and adds their arguments. Repeated multiplication explains why taking a power multiplies the argument.
Unit circle: The circle of radius one centered at the origin in the coordinate plane. Its angle-based coordinates give the geometric meaning of \(\cos\theta+i\sin\theta\).

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