trigonometric formulas from de Moivre identity
When one expands the left hand side of (1) using the binomial theorem (), the sum of the real terms (the real part) must be and the sum of the imaginary terms (cf. the imaginary part) must equal . Thus both and
has been expressed as polynomials of and with integer coefficients.
For example, if , we have
0.1 Linearisation formulas
There are also inverse formulas where one expresses the integer powers and and their products as the polynomials with rational coefficients of either , , … or , , …, depending on whether it is a question of an even (http://planetmath.org/EvenFunction) or an odd function of . We will derive the transformation formulas.
If we denote
then the complex conjugate of is the same as its inverse number:
By adding and subtracting, these equations yield
Similarly, the equations
for any integer . The linearisation formulas are obtained by expanding first the expression to be linearised with the equations (4) and then simplifying the result with the equations (5).
|Title||trigonometric formulas from de Moivre identity|
|Date of creation||2013-03-22 18:51:16|
|Last modified on||2013-03-22 18:51:16|
|Last modified by||pahio (2872)|