what is de moivre's theorem?

Answer 1

de Moivre's formula, named after Abraham de Moivre, states that for any complex number (and, in particular, for any real number) x and any integer n it holds that

(cosx+i sin x)^n = cos (n x) +i*sin(nx)

The formula 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.

Answer 2

De Moivre's Theorem helps us simplify powers of terms with complex numbers easier than expanding
(cosx+isin x)^n = cos(nx)+i*sin(nx)
it can be used to express powers of cos, sin and tan into double angles, triple angles etc
eg can find (sinx)^2 in terms of sin(2x)

Answer 3

(cosx+i sin x)^n = cos (n x) +i*sin(nx)