De Moivre's Formula

De Moivre's formula, also known as de Moivre's theorem, is a result connecting complex numbers and trigonometry. It states that for any real number x and any integer n, the quantity cosine of x plus i times sine of x, raised to the power n, equals cosine of n times x plus i times sine of n times x, where i is the imaginary unit whose square is negative one. The formula is named after the mathematician Abraham de Moivre, although he never stated it explicitly in his own writings. By expanding the left side of the formula and comparing real and imaginary parts, it can be used to derive expressions for the cosine and sine of a multiple angle in terms of the cosine and sine of the original angle, and it can also be used to find the roots of unity, the complex numbers whose nth power equals one.