De Moivre formula
Sine of an angle
Cosine of an angle
Tangent of an angle
Figure 8: Trigonometric functions
A fundamental identity, that links trigonometry with exponential functions,
is the Euler formula
= cos + i sin ,
which expresses a complex number laying on the unit circumference as an ex-
ponential with imaginary exponent
. When is left free to take any real value,
the exponential (23) generates the so-called complex sinusoid.
Any complex number c having magnitude and argument can be repre-
sented in compact form as
c = e
and to it we can apply the usual rules of power functions. For instance, we can
compute the m-th power of c as
(cos m + i sin m) ,
thus showing that it is obtained by taking the m-th power of the magnitude and
multiplying by m the argument. The (25) is called De Moivre formula.
The order-m root of a number c is that number b such that b
= c. In
general, a complex number admits m order-m distinct complex roots
Moivre formula establishes that
the order-m roots of 1 are evenly distributed
The actual meaning of the exponential comes from the series expansion (20)
For instance, 1 admits two square roots (1 and -1) and four order-4 roots (1, -1, i, -i).
The reader is invited to justify this statement by an example. The simplest non-trivial
example is obtained by considering the cubic roots of 1.