de Moivre's theorem
/ duh-MWAHV-ruh /
de Moivre's theorem says that raising a unit-modulus complex number to a power simply multiplies its angle: (cos theta + i sin theta)^n = cos(n theta) + i sin(n theta) for any integer n. In the exponential form this is almost obvious — it is just (e^(i theta))^n = e^(i n theta), the ordinary rule that you multiply exponents. Geometrically, taking the n-th power of a point on the unit circle spins it around n times as fast: the angle is multiplied by n.
More generally, for any z = r e^(i theta), raising to the n-th power gives z^n = r^n e^(i n theta): the modulus is raised to the n-th power and the argument is multiplied by n. So the work of computing z^n, which would be a nightmare of repeated multiplication in rectangular form, becomes a one-step calculation in polar form. To cube 1 + i, for example, write it as (the square root of 2) e^(i pi/4), so its cube is 2^(3/2) e^(i 3 pi/4) = -2 + 2i.
The theorem is a two-way street and that is its real power. Read forwards, it computes high powers in one line. Read backwards, it cracks trigonometry: expanding (cos theta + i sin theta)^n by the binomial theorem and matching real and imaginary parts spills out multiple-angle formulas for cos(n theta) and sin(n theta) for free. And it is the direct stepping stone to extracting n-th roots, where you run the angle-multiplication in reverse.
Compute (1 + i)^8. In polar form 1 + i = (the square root of 2) e^(i pi/4), so (1 + i)^8 = (the square root of 2)^8 e^(i 8 pi/4) = 16 e^(i 2 pi) = 16. A single line, versus eight rounds of multiplying out brackets.
Powers are trivial in polar form: power the modulus, multiply the angle.
Stated for an integer exponent, de Moivre is exact and single-valued. For a fractional exponent like 1/n it 'works' only in the sense of producing one root among several — there are n distinct n-th roots, so you must not treat (cos theta + i sin theta)^(1/n) as a single number.