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Formvlae additionis et svbtractionis, angvli dvpli et dimidii

Svmmarivm

Haec lectio demonstrat formvlas qvae sinvm, cosinvm et tangentem svmmae vel differentiae angvlorvm per rationes vnivscvivsqve angvli exprimvnt. Prima, cosinvs differentiae, obtinetvr longitvdine chordae circvmferentiae vnitariae dvobvs modis compvtata: formvla distantiae in coordinatis Cartesianis et formvla distantiae in coordinatis polaribvs, qvae ostendit chordam a sola differentia angvlorvm pendere. Ex ea, paritate et symmetriis angvlorvm complementarivm adhibitis, dedvcvntvr ceterae formvlae additionis et svbtractionis, et ex his formvlae angvli dvpli, formvlae redvctionis potentiarvm, formvlae angvli dimidii, signo per qvadrantem definito, et formvlae angvli triplicis. His valores exacti compvtantvr, vt ii qvi ad π12\frac{\pi}{12} et π5\frac{\pi}{5} pertinent. Postremo svbstitvtio t=tan⁡θ2t = \tan \frac{\theta}{2} sinvm et cosinvm vt fractiones rationales in tt exprimit, parametrizationem rationalem circvmferentiae vnitariae praebet et, per eam, formvlam qvae omnes ternas Pythagoricas generat.

Proposita discendi

  1. Formvlam cosinvs differentiae per longitvdinem chordae circvmferentiae vnitariae demonstrare, et ex ea formvlas sinvs et tangentis svmmae vel differentiae dedvcere, cvm condicionibvs qvibvs valent.
  2. Formvlas angvli dvpli, redvctionis potentiarvm, angvli dimidii et angvli triplicis dedvcere, et signvm radicis qvadratae secvndvm qvadrantem angvli eligere.
  3. Valores exactos rationvm trigonometricarvm angvlorvm non insignivm compvtare, vt π12\frac{\pi}{12}, 5π12\frac{5\pi}{12} et π5\frac{\pi}{5}, eos ad aeqvationes polynomiales, vbi opvs est, redvcendo.
  4. Sinvm et cosinvm per tangentem angvli dimidii exprimere, parametrizationem rationalem circvmferentiae vnitariae obtinere et per eam omnes ternas Pythagoricas generare.

Cosinvs differentiae

Symmetriae in lectione de rationibvs trigonometricis et circvmferentia vnitaria demonstratae, vt cos⁡(π−θ)=−cos⁡θ\cos(\pi - \theta) = -\cos \theta vel sin⁡(π2−θ)=cos⁡θ\sin\left( \frac{\pi}{2} - \theta \right) = \cos \theta, rationes angvli cvm rationibvs alterivs angvli conivngvnt, qvi angvlo qvadrantali addito vel svbtracto obtinetvr. Iam vero hae aeqvalitates casvs particvlares svnt problematis generalis: rationes ipsorvm α+β\alpha + \beta et α−β\alpha - \beta ex rationibvs ipsorvm α\alpha et β\beta compvtare, qvicvmqve sint hi nvmeri. Ab initio animadvertendvm est rationes in svmmam non distribvi: sin⁡(π6+π3)=sin⁡π2=1\sin\left( \frac{\pi}{6} + \frac{\pi}{3} \right) = \sin \frac{\pi}{2} = 1, cvm sin⁡π6+sin⁡π3=1+32\sin \frac{\pi}{6} + \sin \frac{\pi}{3} = \frac{1 + \sqrt{3}}{2}, qvi maior est qvam 11. Formvla recta sinvs et cosinvs miscet, et clavis ad eam obtinendam est chorda circvmferentiae vnitariae.

Vt in axiomate percvrsvs per longitvdinem arcvs ex lectione de angvlis eorvmqve mensvra, P(θ)P(\theta) significat pvnctvm circvmferentiae vnitariae qvod attingitvr percvrrendo ab (1,0)(1, 0) arcvm longitvdinis ∣θ∣\abs{\theta}, contra motvm horologii si θ>0\theta > 0 et secvndvm motvm horologii si θ<0\theta < 0; ex definitione lectionis citatae, P(θ)=(cos⁡θ,sin⁡θ)P(\theta) = \left( \cos \theta, \sin \theta \right).

TheoremaCosinvs differentiae

Pro omnibvs nvmeris realibvs α\alpha et β\beta valet cos⁡(α−β)=cos⁡αcos⁡β+sin⁡αsin⁡β\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta.

Distantia inter P(α)P(\alpha) et P(β)P(\beta) primvm per coordinatas Cartesianas compvtatvr, deinde per coordinatas polares, qvae svnt (1,α)(1, \alpha) et (1,β)(1, \beta); dvobvs eventibvs aeqvatis formvla obtinetvr.

Demonstratio

  1. d2=(cos⁡α−cos⁡β)2+(sin⁡α−sin⁡β)2d^2 = \left( \resaltar{\cos \alpha - \cos \beta} \right)^2 + \left( \resaltar{\sin \alpha - \sin \beta} \right)^2

    Sit dd distantia inter P(α)P(\alpha) et P(β)P(\beta). Ex formvla distantiae lectionis cvrsvs I de aeqvatione rectae, eivs qvadratvm est svmma qvadratorvm differentiarvm coordinatarvm.

  2. d2=(cos⁡2α+sin⁡2α)+(cos⁡2β+sin⁡2β)−2cos⁡αcos⁡β−2sin⁡αsin⁡β=2−2(cos⁡αcos⁡β+sin⁡αsin⁡β)d^2 = \left( \resaltar{\cos^2 \alpha + \sin^2 \alpha} \right) + \left( \resaltar{\cos^2 \beta + \sin^2 \beta} \right) - 2 \cos \alpha \cos \beta - 2 \sin \alpha \sin \beta = \dato{c}{2 - 2 \left( \cos \alpha \cos \beta + \sin \alpha \sin \beta \right)}

    Dvo qvadrata evolvimvs et sinvm cosinvmqve vnivscvivsqve angvli conivngimvs; ex identitate Pythagorica qvisqve coetvs aeqvat 11.

  3. d2=1+1−2⋅1⋅1⋅cos⁡(α−β)=2−2cos⁡(α−β)d^2 = 1 + 1 - 2 \cdot 1 \cdot 1 \cdot \resaltar{\cos(\alpha - \beta)} = \dato{p}{2 - 2 \cos(\alpha - \beta)}

    Ex theoremate distantiae in coordinatis polaribvs, ex lectione de coordinatis polaribvs, adhibito cvm r1=r2=1r_1 = r_2 = 1, eadem distantia a sola differentia angvlorvm pendet. Geometrice, conversio angvlo −β-\beta circa originem facta dvcit P(β)P(\beta) in (1,0)(1, 0) et P(α)P(\alpha) in P(α−β)P(\alpha - \beta), et distantias servat: dvae chordae figvrae aeqvales svnt.

  4. 2−2cos⁡(α−β)=2−2(cos⁡αcos⁡β+sin⁡αsin⁡β)↔cos⁡(α−β)=cos⁡αcos⁡β+sin⁡αsin⁡β\resaltar{2 - 2 \cos(\alpha - \beta)} = \resaltar{2 - 2 \left( \cos \alpha \cos \beta + \sin \alpha \sin \beta \right)} \leftrightarrow \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta

    Dvas expressiones ipsivs d2d^2 aeqvamvs, 22 svbtrahimvs et per −2-2 dividimvs.

Demonstratio nvllam condicionem de α\alpha et β\beta postvlat: theorema distantiae in coordinatis polaribvs pro qvibvslibet angvlis valet, et si P(α)=P(β)P(\alpha) = P(\beta) ambae expressiones ipsivs d2d^2 nvllae svnt. Neqve est circvlvs vitiosvs: illvd theorema theoremate cosinvvm demonstratvm est, qvod ipsvm obtentvm est triangvlo in systemate coordinatarvm collocato, nvllis formvlis additionis angvlorvm adhibitis. Via etiam retro percvrri potest: si latera bb et cc triangvli ab origine cvm inclinationibvs β\beta et α\alpha exevnt, qvadratvm distantiae inter eorvm extrema est b2+c2−2bc(cos⁡αcos⁡β+sin⁡αsin⁡β)b^2 + c^2 - 2bc \left( \cos \alpha \cos \beta + \sin \alpha \sin \beta \right), et qvi formvlam cosinvs differentiae habet, ex eo theorema cosinvvm lectionis de solvtione triangvlorvm obtinet.