De Moivre's Theorem: (cos theta + i sin theta)^n = cos(ntheta) + i sin(ntheta)

Applications:

1. Computing powers of complex numbers: Convert to polar form, apply De Moivre, convert back. Example: (1+i)^10 = (sqrt(2))^10 * (cos(pi/4)+i sin(pi/4))^10 = 32*(cos(5pi/2)+i sin(5pi/2)) = 32*(0+i) = 32i

2. Finding nth roots: z^n = w has n roots: z_k = |w|^(1/n) * e^(i*(arg(w)+2k*pi)/n), k = 0, ..., n-1

• All roots have the same modulus: |w|^(1/n)
• Roots are equally spaced on a circle, separated by 2*pi/n
• They form a regular n-gon

3. Trigonometric identities: Expand (cos theta + i sin theta)^n using binomial theorem, then equate real parts to get cos(ntheta) and imaginary parts to get sin(ntheta). Example: cos(3theta) = cos^3(theta) - 3cos(theta)sin^2(theta)

4. Summation of trigonometric series: cos(a) + cos(a+d) + ... + cos(a+(n-1)d) = Re(sum of geometric series in e^(i*theta))

Key identities from De Moivre:

• z + 1/z = 2cos theta (when |z|=1)
• z - 1/z = 2i sin theta
• z^n + 1/z^n = 2cos(n*theta)