trigonometric formulas from de Moivre identity

De Moivre identityMathworldPlanetmath

(cosφ+isinφ)n=cosnφ+isinnφ  (n) (1)

implies simply some important trigonometric formulas, the derivation of which without imaginary numbersMathworldPlanetmath would require much longer calculations.

When one expands the left hand side of (1) using the binomial theorem (n>0), the sum of the real terms (the real partDlmfMathworldPlanetmath) must be cosnφ and the sum of the imaginary terms (cf. the imaginary part) must equal isinnφ.  Thus both cosnφ and sinnφ has been expressed as polynomialsPlanetmathPlanetmath of sinφ and cosφ with integer coefficients.

For example, if  n=5,  we have



sin5φ= 5cos4φsinφ-10cos2φsin3φ+sin5φ.

By the “fundamental formula”  sin2φ+cos2φ=1  of trigonometry, the even powers on the right hand sides may be expressed with the other functionMathworldPlanetmath; therefore we obtain

cos5φ= 16cos5φ-20cos3φ+5cosφ, (2)
sin5φ= 16sin5φ-20sin3φ+5sinφ. (3)

0.1 Linearisation formulas

There are also inverse formulas where one expresses the integer powers cosmφ and sinnφ and their products as the polynomials with rational coefficients of either cosφ, cos2φ, …  or sinφ, sin2φ, …,  depending on whether it is a question of an even ( or an odd functionMathworldPlanetmath of φ.  We will derive the transformation formulas.

If we denote


then the complex conjugateDlmfMathworldPlanetmath of t is the same as its inverse number:


By adding and subtracting, these equations yield

cosφ=12(t+1t),sinφ=12i(t-1t). (4)

Similarly, the equations



cosnφ=12(tn+1tn),sinnφ=12i(tn-1tn). (5)

for any integer n.  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).

Example 1.

cos4φ =(12(t+1t))4

Example 2.

cos4φsin3φ =116(t+1t)4-18i(t-1t)3
Title trigonometric formulas from de Moivre identity
Canonical name TrigonometricFormulasFromDeMoivreIdentity
Date of creation 2013-03-22 18:51:16
Last modified on 2013-03-22 18:51:16
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 16
Author pahio (2872)
Entry type Derivation
Classification msc 30D05
Classification msc 30A99
Related topic TrigonometricFormulasFromSeries
Related topic ReductionFormulas
Related topic GoniometricFormulae