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Transformatio svmmarvm in prodvcta et forma harmonica

Svmmarivm

Haec lectio ex formvlis sinvs et cosinvs svmmae et differentiae eas dedvcit qvae prodvcta sinvvm et cosinvvm in svmmas et, vicissim, svmmas in prodvcta convertvnt, et has posteriores in circvmferentia vnitaria per pvnctvm medivm chordae interpretatvr. Demonstrat omnem expressionem asin⁡x+bcos⁡xa \sin x + b \cos x in forma harmonica Rsin⁡(x+φ)R \sin(x + \varphi) scribi, cvm R=a2+b2R = \sqrt{a^2 + b^2} et angvlo φ\varphi vnico in [0,2π)\ico{0}{2\pi}, inde copiam imaginvm eivs dvabvs viis obtinet, qvarvm altera est inaeqvalitas Cavchy-Schwarziana, et conclvdit svperpositionem dvarvm oscillationvm eivsdem freqventiae esse rvrsvs oscillationem. Pvlsationes explicat vt prodvctvm oscillationis lentae et celeris et, mvltiplicando per factorem qvi svmmam telescopicam reddit, svmmas sinvvm et cosinvvm angvlorvm in progressione arithmetica forma clavsa compvtat.

Proposita discendi

  1. Formvlas de prodvcto in svmmam et de svmma in prodvctvm ex formvlis additionis et svbtractionis dedvcere, et has posteriores in circvmferentia vnitaria interpretari.
  2. Omnem expressionem asin⁡x+bcos⁡xa \sin x + b \cos x in forma harmonica scribere, existentiam et vnicitatem amplitvdinis et angvli eivs demonstrare, eaqve vti ad copias imaginvm, maxima et minima invenienda.
  3. Svperpositionem dvarvm oscillationvm eivsdem freqventiae vt oscillationem agnoscere, et pvlsationes per involvcrvm earvm describere.
  4. Svmmas sinvvm et cosinvvm in progressione arithmetica per svmmas telescopicas compvtare, et has transformationes ad expressiones simplificandas et ad identitates demonstrandas adhibere.

A prodvctis ad svmmas

In lectione de formvlis additionis et svbtractionis demonstratae svnt, pro omnibvs realibvs α\alpha et β\beta, qvattvor aeqvalitates infra scriptae:

sin⁡(α+β)=sin⁡αcos⁡β+cos⁡αsin⁡β,sin⁡(α−β)=sin⁡αcos⁡β−cos⁡αsin⁡β\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta, \qquad \sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta
cos⁡(α+β)=cos⁡αcos⁡β−sin⁡αsin⁡β,cos⁡(α−β)=cos⁡αcos⁡β+sin⁡αsin⁡β\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta, \qquad \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta

Dvae formvlae cvivsqve paris eadem dvo prodvcta continent, et solo signo secvndi differvnt. Qvod si ita est, cvm addvntvr avt svbtrahvntvr, altervm prodvctvm evanescit et altervm dvplicatvr: qvodqve prodvctvm sinvs avt cosinvs per alivm vt semisvmma avt semidifferentia sinvvm avt cosinvvm exprimitvr. Hae formvlae, prosthaphaeresis appellatae (e Graeco, «additio et svbtractio»), astronomis exevnte saecvlo XVI, in qvibvs Tycho Brahe, vsvi fvervnt ad mvltiplicationem nvmerorvm mvltorvm digitorvm ad additiones et svbtractiones redigendam, tabvlis sinvvm consvltis, anteqvam Ioannes Napiervs anno 1614 tabvlas logarithmorvm svas ederet.

TheoremaDe prodvcto in svmmam

Pro omnibvs realibvs α\alpha et β\beta, prodvctvm sin⁡αcos⁡β\sin \alpha \cos \beta est semisvmma ipsorvm sin⁡(α+β)\sin(\alpha + \beta) et sin⁡(α−β)\sin(\alpha - \beta); prodvctvm cos⁡αsin⁡β\cos \alpha \sin \beta est semidifferentia eorvndem sinvvm; prodvctvm cos⁡αcos⁡β\cos \alpha \cos \beta est semisvmma ipsorvm cos⁡(α−β)\cos(\alpha - \beta) et cos⁡(α+β)\cos(\alpha + \beta), et prodvctvm sin⁡αsin⁡β\sin \alpha \sin \beta est eorvm semidifferentia.

sin⁡αcos⁡β=12(sin⁡(α+β)+sin⁡(α−β)),cos⁡αsin⁡β=12(sin⁡(α+β)−sin⁡(α−β))\sin \alpha \cos \beta = \frac{1}{2} \left( \sin(\alpha + \beta) + \sin(\alpha - \beta) \right), \qquad \cos \alpha \sin \beta = \frac{1}{2} \left( \sin(\alpha + \beta) - \sin(\alpha - \beta) \right)
cos⁡αcos⁡β=12(cos⁡(α−β)+cos⁡(α+β)),sin⁡αsin⁡β=12(cos⁡(α−β)−cos⁡(α+β))\cos \alpha \cos \beta = \frac{1}{2} \left( \cos(\alpha - \beta) + \cos(\alpha + \beta) \right), \qquad \sin \alpha \sin \beta = \frac{1}{2} \left( \cos(\alpha - \beta) - \cos(\alpha + \beta) \right)

Qvaeqve formvla oritvr ex dvabvs formvlis paris addendis avt svbtrahendis. Figvra eas illvstrat cvm α=2x\alpha = 2x et β=x\beta = x: prodvctvm, linea crassa dvctvm, congrvit cvm semisvmma avt semidifferentia dvarvm oscillationvm freqventiarvm 33 et 11, linea tenvi dvctarvm.

Demonstratio

  1. sin⁡(α+β)+sin⁡(α−β)=2sin⁡αcos⁡β→sin⁡αcos⁡β=12(sin⁡(α+β)+sin⁡(α−β))\sin(\alpha + \beta) + \sin(\alpha - \beta) = \resaltar{2 \sin \alpha \cos \beta} \rightarrow \dato{sc}{\sin \alpha \cos \beta = \frac{1}{2} \left( \sin(\alpha + \beta) + \sin(\alpha - \beta) \right)}

    Formvlas sinvs svmmae et sinvs differentiae addimvs: termini cos⁡αsin⁡β\cos \alpha \sin \beta evanescvnt. Dividendo per 22 prima formvla obtinetvr.

  2. sin⁡(α+β)−sin⁡(α−β)=2cos⁡αsin⁡β→cos⁡αsin⁡β=12(sin⁡(α+β)−sin⁡(α−β))\sin(\alpha + \beta) - \sin(\alpha - \beta) = \resaltar{2 \cos \alpha \sin \beta} \rightarrow \cos \alpha \sin \beta = \frac{1}{2} \left( \sin(\alpha + \beta) - \sin(\alpha - \beta) \right)

    Si eas pro addendo svbtrahimvs, termini sin⁡αcos⁡β\sin \alpha \cos \beta evanescvnt.

  3. cos⁡(α−β)+cos⁡(α+β)=2cos⁡αcos⁡β→cos⁡αcos⁡β=12(cos⁡(α−β)+cos⁡(α+β))\cos(\alpha - \beta) + \cos(\alpha + \beta) = \resaltar{2 \cos \alpha \cos \beta} \rightarrow \dato{cc}{\cos \alpha \cos \beta = \frac{1}{2} \left( \cos(\alpha - \beta) + \cos(\alpha + \beta) \right)}

    Cvm formvlis cosinvs eodem modo agitvr. Cvm addvntvr, prodvcta sinvvm evanescvnt.

  4. cos⁡(α−β)−cos⁡(α+β)=2sin⁡αsin⁡β→sin⁡αsin⁡β=12(cos⁡(α−β)−cos⁡(α+β))\cos(\alpha - \beta) - \cos(\alpha + \beta) = \resaltar{2 \sin \alpha \sin \beta} \rightarrow \dato{ss}{\sin \alpha \sin \beta = \frac{1}{2} \left( \cos(\alpha - \beta) - \cos(\alpha + \beta) \right)}

    Cvm formvla cosinvs svmmae a formvla cosinvs differentiae svbtrahitvr, prodvcta cosinvvm evanescvnt.

  5. sin⁡αcos⁡α=12sin⁡(2α),cos⁡2α=12(1+cos⁡(2α)),sin⁡2α=12(1−cos⁡(2α))\sin \alpha \cos \alpha = \frac{1}{2} \sin(2\alpha), \quad \cos^2 \alpha = \frac{1}{2} \left( 1 + \cos(2\alpha) \right), \quad \sin^2 \alpha = \frac{1}{2} \left( 1 - \cos(2\alpha) \right)

    Cvm β=α\beta = \alpha, et qvia sin⁡0=0\sin 0 = 0 et cos⁡0=1\cos 0 = 1, recvperantvr formvla angvli dvpli sinvs et formvlae redvctionis potentiarvm ex lectione de formvlis additionis et svbtractionis. In figvra, sin⁡2x\sin^2 x periodo π\pi circa 12\frac{1}{2} oscillat.

Formvlae sinvnt prodvcta compvtare qvae ad valores insignes factore per factorem non redigvntvr. Exempli gratia, cvm α=5π12\alpha = \frac{5\pi}{12} et β=π12\beta = \frac{\pi}{12}, svmma angvlorvm est π2\frac{\pi}{2} et differentia eorvm est π3\frac{\pi}{3}:

cos⁡5π12cos⁡π12=12(cos⁡π3+cos⁡π2)=12(12+0)=14\cos \frac{5\pi}{12} \cos \frac{\pi}{12} = \frac{1}{2} \left( \cos \frac{\pi}{3} + \cos \frac{\pi}{2} \right) = \frac{1}{2} \left( \frac{1}{2} + 0 \right) = \frac{1}{4}

Attamen praecipva harvm formvlarvm vtilitas alia est: prodvctvm dvarvm oscillationvm, qvod difficile cogitatv est, est svperpositio dvarvm oscillationvm qvarvm freqventiae svnt svmma et differentia freqventiarvm factorvm. Sic sin⁡(3x)cos⁡x=12(sin⁡(4x)+sin⁡(2x))\sin(3x) \cos x = \frac{1}{2} \left( \sin(4x) + \sin(2x) \right).