De Moivre's formula

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. 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. For and , de Moivre's formula asserts that or equivalently that In this example, it is easy to check the validity of the equation by multiplying out the left side. De Moivre's formula is a precursor to Euler's formula which establishes the fundamental relationship between the trigonometric functions and the complex exponential function. One can derive de Moivre's formula using Euler's formula and the exponential law for integer powers since Euler's formula implies that the left side is equal to while the right side is equal to The truth of de Moivre's theorem can be established by using mathematical induction for natural numbers, and extended to all integers from there. For an integer n, call the following statement S(n): For n > 0, we proceed by mathematical induction. S(1) is clearly true. For our hypothesis, we assume S(k) is true for some natural k. That is, we assume Now, considering S(k + 1): See angle sum and difference identities. We deduce that S(k) implies S(k + 1). By the principle of mathematical induction it follows that the result is true for all natural numbers. Now, S(0) is clearly true since cos(0x) + i sin(0x) = 1 + 0i = 1.