Imaginary and complex numbers Exponentiation and root extraction of complex numbers in the polar form Euler’s formula, relationship between trigonometric functions and the complex exponential function
Exponentiation and root extraction of complex numbers in the polar form - de Moivre's formula
We use the polar form for exponentiation and root extraction of complex numbers that are known as de Moivre's formulas.
 zn = rn · [cos(nj) + isin(nj)]
 and Powers and roots of complex numbers, use of de Moivre’s formulas examples
 Example: Compute Solution: or in the polar form, and since exponentiation with integer exponent then Example: Compute Solution:  As square root of a complex number is a complex number, then and, two complex numbers are equal if their real parts are equal and their imaginary parts are equal, that is  Example: Calculate using de Moivre's formula.
 Solution:  since then   These complex numbers satisfy the equation z3 = -8 and by the Fundamental theorem of algebra, since this equation is of degree 3, there must be 3 roots.
Thus, for example to check the root zk=2 we cube this solution, then Example: Calculate Solution:     Euler’s formula, relationship between trigonometric functions and the complex exponential function
 Euler's formula eij = cosj + isinj,  where e is the base of the natural logarithm, i is the imaginary unit, and j is the angle between x-axis and the vector pointing to the complex number z measured counter clockwise, that is, j is the argument of z, describes the unit circle in the complex plane. That is, on the unit circle lie points of the complex plane that correspond to the complex numbers each of which is one unit far from the origin.
 Thus, by plugging the angles, into Euler's formula respectively obtained are the four complex numbers that lie on the unit circle, the two of which lie on the real axis and the two on the imaginary axis as shows the above picture.
The expression  eip + 1 = 0  is called Euler's equation or identity.
Euler's formula shows relations between trigonometric functions and complex exponentials.
Thus, by adding and subtracting Euler's formulas:
(1)   eij = cosj + isinj
and                              (2)   e-ij = cos(-j) + isin(-j) = cosj - isinj
 and solving for cosine and sine We use Euler's formula to write complex number z = x + yi in polar coordinates,
 z = x + yi = r (cosj + isinj) = r eij
 where     College algebra contents 