Cue Column:

• State De Moivre's theorem
• How to find nth roots?
• How to expand cos(n*theta)?

Note Column: Theorem: (cos theta + i sin theta)^n = cos(ntheta) + i sin(ntheta) for all integers n.

Application 1: nth roots of a complex number The n roots of $z^n$ = w (where w = Re^(iphi)) are: $z_k$ = R^$\frac{1}{n}$ * e^$\frac{i*(phi + 2k*pi}{n}$) for k = 0, 1, ..., n-1

These roots are equally spaced on a circle of radius R^$\frac{1}{n}$, separated by angle 2*pi/n.

Application 2: Trigonometric identities Expand (cos theta + i sin theta)^n using binomial theorem, then compare real and imaginary parts to get expressions for cos(ntheta) and sin(ntheta) as polynomials in cos theta and sin theta.

Application 3: Summation of series cos(alpha) + cos(alpha+beta) + ... + cos(alpha+(n-1)*beta) = Re(geometric series of complex exponentials).

Summary: De Moivre converts between powers and multiple angles. The n roots are uniformly distributed on a circle.