## De Moivre's Theorem

For complex number z and any integer n, De Moivre's theorem states:

z

^{n}= (cos x + i sin x)^{n}= cos(nx) + i sin(nx)The theorem is important because it connects complex numbers (i stands for the imaginary unit) and trigonometry. The expression "cos x + i sin x" is sometimes abbreviated to "cis x".

By expanding the left hand side and then comparing the real and imaginary parts, it is possible to derive useful expressions for cos(nx) and sin(nx) in terms of cos(x) and sin(x). Furthermore, one can use this formula to find explicit expressions for the n-th roots of unity, that is, complex numbers z such that zn = 1.