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trigonometric formulas from de Moivre identity

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trigonometric formulas from de Moivre identity

De Moivre identity

(cosφ+isinφ)n=cosnφ+isinnφ  (n∈ℤ)fragmentssuperscriptfragmentsnormal-(φiφnormal-)nnφinφitalic-  fragmentsnormal-(nZnormal-)\displaystyle(\cos\varphi+i\sin\varphi)^{n}\,=\,\cos{n\varphi}+i\sin{n\varphi}% \qquad(n\in\mathbb{Z}) (1)

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

When one expands the left hand side of (1) using the binomial theorem (n>0n0n>0), the sum of the real terms (the real part) must be cos⁡n⁢φnφ\cos{n\varphi} and the sum of the imaginary terms (cf. the imaginary part) must equal i⁢sin⁡n⁢φinφi\sin{n\varphi}.  Thus both cos⁡n⁢φnφ\cos{n\varphi} and sin⁡n⁢φnφ\sin{n\varphi} has been expressed as polynomials of sin⁡φφ\sin\varphi and cos⁡φφ\cos\varphi with integer coefficients.

For example, if  n=5n5n=5,  we have

(cos⁡φ+i⁢sin⁡φ)5=cos5⁡φ+5⁢i⁢cos4⁡φ⁢sin⁡φ-10⁢cos3⁡φ⁢sin2⁡φ-10⁢i⁢cos2⁡φ⁢sin3⁡φ+5⁢cos⁡φ⁢sin4⁡φ+i⁢sin5⁡φ,superscriptφiφ5superscript5φ5isuperscript4φφ10superscript3φsuperscript2φ10isuperscript2φsuperscript3φ5φsuperscript4φisuperscript5φ(\cos\varphi+i\sin\varphi)^{5}\;=\;\cos^{5}\varphi+5i\cos^{4}\varphi\sin% \varphi-10\cos^{3}\varphi\sin^{2}\varphi-10i\cos^{2}\varphi\sin^{3}\varphi+5% \cos\varphi\sin^{4}\varphi+i\sin^{5}\varphi,

whence

cos⁡5⁢φ=cos5⁡φ-10⁢cos3⁡φ⁢sin2⁡φ+5⁢cos⁡φ⁢sin4⁡φ,5φsuperscript5φ10superscript3φsuperscript2φ5φsuperscript4φ\cos{5\varphi}\;=\;\cos^{5}\varphi-10\cos^{3}\varphi\sin^{2}\varphi+5\cos% \varphi\sin^{4}\varphi,
sin⁡5⁢φ= 5⁢cos4⁡φ⁢sin⁡φ-10⁢cos2⁡φ⁢sin3⁡φ+sin5⁡φ.5φ 5superscript4φφ10superscript2φsuperscript3φsuperscript5φ\sin{5\varphi}\;=\;5\cos^{4}\varphi\sin\varphi-10\cos^{2}\varphi\sin^{3}% \varphi+\sin^{5}\varphi.

By the “fundamental formula”  sin2⁡φ+cos2⁡φ=1superscript2φsuperscript2φ1\sin^{2}\varphi+\cos^{2}\varphi=1  of trigonometry, the even powers on the right hand sides may be expressed with the other function; therefore we obtain

cos⁡5⁢φ= 16⁢cos5⁡φ-20⁢cos3⁡φ+5⁢cos⁡φ,5φ 16superscript5φ20superscript3φ5φ\displaystyle\cos{5\varphi}\;=\;16\cos^{5}\varphi-20\cos^{3}\varphi+5\cos\varphi, (2)
sin⁡5⁢φ= 16⁢sin5⁡φ-20⁢sin3⁡φ+5⁢sin⁡φ.5φ 16superscript5φ20superscript3φ5φ\displaystyle\sin{5\varphi}\;=\;16\sin^{5}\varphi-20\sin^{3}\varphi+5\sin\varphi. (3)

0.1 Linearisation formulas

There are also inverse formulas where one expresses the integer powers cosm⁡φsuperscriptmφ\cos^{m}\varphi and sinn⁡φsuperscriptnφ\sin^{n}\varphi and their products as the polynomials with rational coefficients of either cos⁡φφ\cos\varphi, cos⁡2⁢φ2φ\cos 2\varphi, …  or sin⁡φφ\sin\varphi, sin⁡2⁢φ2φ\sin 2\varphi, …,  depending on whether it is a question of an even or an odd function of φφ\varphi.  We will derive the transformation formulas.

If we denote

cos⁡φ+i⁢sin⁡φ:=t,assignφiφt\cos\varphi+i\sin\varphi\;:=\;t,

then the complex conjugate of ttt is the same as its inverse number:

cos⁡φ-i⁢sin⁡φ=1t.φiφ1t\cos\varphi-i\sin\varphi\;=\;\frac{1}{t}.

By adding and subtracting, these equations yield

cos⁡φ=12⁢(t+1t),sin⁡φ=12⁢i⁢(t-1t).formulae-sequenceφ12t1tφ12it1t\displaystyle\cos\varphi\;=\;\frac{1}{2}\left(t+\frac{1}{t}\right),\quad\sin% \varphi\;=\;\frac{1}{2i}\left(t-\frac{1}{t}\right). (4)

Similarly, the equations

(cos⁡φ+i⁢sin⁡φ)±n=cos⁡(±n⁢φ)+i⁢sin⁡(±n⁢φ)superscriptφiφplus-or-minusnplus-or-minusnφiplus-or-minusnφ(\cos\varphi+i\sin\varphi)^{{\pm n}}\,=\,\cos(\pm n\varphi)+i\sin(\pm n\varphi)

yield

cos⁡n⁢φ=12⁢(tn+1tn),sin⁡n⁢φ=12⁢i⁢(tn-1tn).formulae-sequencenφ12superscripttn1superscripttnnφ12isuperscripttn1superscripttn\displaystyle\cos{n\varphi}\;=\;\frac{1}{2}\left(t^{n}+\frac{1}{t^{n}}\right),% \quad\sin{n\varphi}\;=\;\frac{1}{2i}\left(t^{n}-\frac{1}{t^{n}}\right). (5)

for any integer nnn.  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.

cos4⁡φsuperscript4φ\displaystyle\cos^{4}\varphi =(12⁢(t+1t))4absentsuperscript12t1t4\displaystyle\;=\;\left(\frac{1}{2}\left(t+\frac{1}{t}\right)\right)^{4}
=116⁢(t4+4⁢t2+6+4t2+1t4)absent116superscriptt44superscriptt264superscriptt21superscriptt4\displaystyle\;=\;\frac{1}{16}\left(t^{4}+4t^{2}+6+\frac{4}{t^{2}}+\frac{1}{t^% {4}}\right)
=116⁢(t4+1t4)+14⁢(t2+1t2)+38absent116superscriptt41superscriptt414superscriptt21superscriptt238\displaystyle\;=\;\frac{1}{16}\left(t^{4}+\frac{1}{t^{4}}\right)+\frac{1}{4}% \left(t^{2}+\frac{1}{t^{2}}\right)+\frac{3}{8}
=18⁢cos⁡4⁢φ+12⁢cos⁡2⁢φ+38absent184φ122φ38\displaystyle\;=\;\frac{1}{8}\cos 4\varphi+\frac{1}{2}\cos 2\varphi+\frac{3}{8}

Example 2.

cos4⁡φ⁢sin3⁡φsuperscript4φsuperscript3φ\displaystyle\cos^{4}\varphi\sin^{3}\varphi =116⁢(t+1t)4⁢-18⁢i⁢(t-1t)3absent116superscriptt1t418isuperscriptt1t3\displaystyle\;=\;\frac{1}{16}\left(t+\frac{1}{t}\right)^{4}\frac{-1}{8i}\left% (t-\frac{1}{t}\right)^{3}
=-1128⁢i⁢(t2-1t2)3⁢(t+1t)absent1128isuperscriptsuperscriptt21superscriptt23t1t\displaystyle\;=\;-\frac{1}{128i}\left(t^{2}-\frac{1}{t^{2}}\right)^{3}\left(t% +\frac{1}{t}\right)
=-1128⁢i⁢(t6-3⁢t2+3t2-1t6)⁢(t+1t)absent1128isuperscriptt63superscriptt23superscriptt21superscriptt6t1t\displaystyle\;=\;-\frac{1}{128i}\left(t^{6}-3t^{2}+\frac{3}{t^{2}}-\frac{1}{t% ^{6}}\right)\left(t+\frac{1}{t}\right)
=-1128⁢i⁢(t7-3⁢t3+3t-1t5-3⁢t+3t3-1t7)absent1128isuperscriptt73superscriptt33t1superscriptt53t3superscriptt31superscriptt7\displaystyle\;=\;-\frac{1}{128i}\left(t^{7}-3t^{3}+\frac{3}{t}-\frac{1}{t^{5}% }-3t+\frac{3}{t^{3}}-\frac{1}{t^{7}}\right)
=-1128⁢i⁢((t7-1t7)+(t5-1t5)-3⁢(t3-1t3)-3⁢(t-1t))absent1128isuperscriptt71superscriptt7superscriptt51superscriptt53superscriptt31superscriptt33t1t\displaystyle\;=\;-\frac{1}{128i}\left(\left(t^{7}-\frac{1}{t^{7}}\right)+% \left(t^{5}-\frac{1}{t^{5}}\right)-3\left(t^{3}-\frac{1}{t^{3}}\right)-3\left(% t-\frac{1}{t}\right)\right)
=-164⁢sin⁡7⁢φ-164⁢sin⁡5⁢φ+364⁢sin⁡3⁢φ+364⁢sin⁡φabsent1647φ1645φ3643φ364φ\displaystyle\;=\;-\frac{1}{64}\sin 7\varphi-\frac{1}{64}\sin 5\varphi+\frac{3% }{64}\sin 3\varphi+\frac{3}{64}\sin\varphi
Related: 
TrigonometricFormulasFromSeries, ReductionFormulas, GoniometricFormulae
Major Section: 
Reference
Type of Math Object: 
Derivation
Parent: 
de Moivre identity

Mathematics Subject Classification

30D05 Functional equations in the complex domain, iteration and composition of analytic functions
30A99 None of the above, but in MSC2010 section 30Axx

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Owner: pahio
Added: 2009-03-16 - 01:45
Author(s): pahio

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