An Automorphism of de Moivre’s Theorem (Math 347)

Introduction to de Moivre’s Theorem:
https://brilliant.org/wiki/de-moivres-theorem/

Practice Problem:

Books: PreCalculus text, Dummit and Foote

Preliminary Reading: https://www.wikiwand.com/en/Abraham_de_Moivre

Abraham de Moivre (French pronunciation: ​[abʁaam də mwavʁ]; 26 May 1667 – 27 November 1754) was a French mathematician known for de Moivre’s formula, a formula that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory.

In mathematics, de Moivre’s formula (also known as de Moivre’s theorem and de Moivre’s identity) states that for any real number x and integer n it holds that

where i is the imaginary unit (i² = −1). The formula is named after Abraham de Moivre, although he never stated it in his works.^[1] The expression cos x + i sin x is sometimes abbreviated to cis x.

The formula is important because it connects complex numbers and trigonometry. By expanding the left hand side and then comparing the real and imaginary parts under the assumption that x is real, it is possible to derive useful expressions for cos nx and sin nx in terms of cos x and sin x.

As written, the formula is not valid for non-integer powers n. However, there are generalizations of this formula valid for other exponents. These can be used to give explicit expressions for the nth roots of unity, that is, complex numbers z such that zⁿ = 1.