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Trigonometria analytica: synthesis et provocationes

Svmmarivm

Haec lectio partem de trigonometria analytica clavdit. In vnam catenam dedvctivam colligit qvae in sex eivs lectionibvs demonstrata svnt: identitates fvndamentales, formvlam cosinvs differentiae omnesqve qvae ex ea dedvcvntvr, transformationem svmmarvm in prodvcta et formam harmonicam, solvtionem aeqvationvm et inaeqvationvm, angvlvm dvarvm rectarvm et rotationem axivm qvae sectiones conicas classificat. Explicat qvomodo haec catena parti antecedenti et cvrsvi Algebra et Geometria I innitatvr, praesertim circvmferentiae vnitariae, coordinatis polaribvs, polynomiis et sectionibvs conicis, et qvomodo partem qvae seqvitvr praeparet, in qva eaedem formvlae vt regvla mvltiplicationis nvmerorvm complexorvm redevnt. Probatione finali finitvr, qvae probationi initiali introdvctionis respondet, et decem provocationibvs, difficilioribvs qvam problemata partis proposita, qvae plvres lectiones conivngere postvlant.

Proposita discendi

  1. Catenam dedvctivam partis restitvere, a formvla cosinvs differentiae vsqve ad classificationem sectionvm conicarvm per discriminantem, et in qvoqve nexv effectvm agnoscere a qvo pendet.
  2. Effectvs partis cvm effectibvs partis antecedentis et cvrsvs I de circvmferentia vnitaria, coordinatis polaribvs, polynomiis et sectionibvs conicis conferre.
  3. Mvnvs formvlarvm additionis et svbtractionis in mvltiplicatione nvmerorvm complexorvm forma polari praevidere.
  4. Problemata solvere qvae identitates, aeqvationes, angvlos rectarvm et rotationes conivngvnt, in singvlis eam transformationem eligendo qvae problema tractabile reddit.

Qvae vnitas consecvta est

Pars a lingva statvenda incepit. In lectione de identitatibvs trigonometricis fvndamentalibvs identitas definita est vt aeqvalitas qvae in toto dominio commvni dvorvm membrorvm valet, identitates reciprocae, qvotientis, Pythagoricae, paritatis et cofvnctionis collectae svnt, et methodvs demonstrandi statvta est: altervm membrvm in altervm transformare, avt vtrvmqve in eandem expressionem, non ab eo proficiscendo qvod demonstrandvm est; vnvm exemplvm contrarivm svfficit vt identitas falsa refvtetvr. Hac lingva qvaeqve fvnctio per aliam expressa est, radices per svbstitvtiones transformatae svnt, et ellipsis atqve hyperbola in parametrizationibvs (acos⁡t,bsin⁡t)\left( a \cos t, b \sin t \right) et (asec⁡t,btan⁡t)\left( a \sec t, b \tan t \right) agnitae svnt.

Momentvm decretorivm fvit lectio de formvlis additionis et svbtractionis. Longitvdine chordae circvmferentiae vnitariae dvobvs modis compvtata, in coordinatis Cartesianis et in coordinatis polaribvs, demonstratvm est pro omnibvs realibvs α\alpha et β\beta aeqvalitatem infra scriptam valere:

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

Cetera omnia ex ea dedvcta svnt. Paritate cosinvs svmmae obtentvs est; cofvnctione, sinvs svmmae et differentiae; qvotiente, tangens svmmae. Qvod si ita est, posito α=β\alpha = \beta formvlae angvli dvpli obtinentvr; qvadratis segregatis, formvlae redvctionis potentiarvm et angvli dimidii; itervm addendo, formvlae angvli triplicis. Svbstitvtio t=tan⁡θ2t = \tan \frac{\theta}{2} sinvm et cosinvm in fractiones rationales ipsivs tt convertit, et inde parametrizatio rationalis circvmferentiae et formvla omnivm ternarvm Pythagoricarvm secvtae svnt.

Lectio de transformatione svmmarvm in prodvcta viam contrariam percvrrit: formvlis additionis et svbtractionis addendis et svbtrahendis prodvcta in svmmas et svmmae in prodvcta convertvntvr. Eivs effectvs praecipvvs est omnem combinationem asin⁡x+bcos⁡xa \sin x + b \cos x vnam vndam Rsin⁡(x+φ)R \sin(x + \varphi) esse, cvm R=a2+b2R = \sqrt{a^2 + b^2}; inde seqvvntvr eivs copia imaginvm, [−R,R]\icc{-R}{R}, svperpositio oscillationvm eivsdem freqventiae, pvlsationes et, factore qvi svmmam telescopicam reddit, svmmae sinvvm et cosinvvm in progressione arithmetica.

His instrvmentis lectio de aeqvationibvs et inaeqvationibvs trigonometricis totam problematvm familiam solvere potvit. Solvtiones generales ipsarvm sin⁡x=c\sin x = c, cos⁡x=c\cos x = c et tan⁡x=c\tan x = c eas qvas fvnctiones inversae dant per periodicitatem extendvnt; ceterae aeqvationes ad has redigvntvr per identitates, factorizationem, aeqvalitatem dvorvm sinvvm avt dvorvm cosinvvm et formam harmonicam, dvabvs cavtionibvs semper servatis: dividendo solvtiones amitti possvnt, et qvadrando addi. Inaeqvationes in vna conversione solvtae svnt, circvmferentia vnitaria avt tabvlis signorvm, et per periodicitatem extensae.

Dvae vltimae lectiones hvnc calcvlvm in geometriam analyticam transtvlervnt. In lectione de angvlo dvarvm rectarvm tangens differentiae angvlvm dvarvm rectarvm ex earvm declivitatibvs dedit, et prodvctvm scalare alteram formvlam, rectis verticalibvs qvoqve validam; bisectrices et incentrvm obtenta svnt, et altitvdines inaccessae, motvs harmonicvs, spativm iactvs proiectilivm et reflexio lvcis ad formam mathematicam redacta svnt. Illa lectio sola partis aliqvid sine demonstratione admittit, idqve declarat: ex geometria Evclidea tria facta qvae cvrsvs I iam admittebat (theorema Pythagorae, aeqvalitatem angvlorvm correspondentivm inter parallelas et svmmam π\pi angvlorvm triangvli), et ex physica aeqvationes motvs harmonici simplicis et proiectilis. Postremo, in lectione de rotatione axivm, formvlae differentiae mvtationem coordinatarvm axibvs rotatis descripservnt; angvlvs qvi cot⁡2θ=A−CB\cot 2\theta = \frac{A - C}{B} implet terminvm in xyxy eliminat, svmma A+CA + C et discriminans B2−4ACB^2 - 4AC rotatione non mvtantvr, ideoqve signvm discriminantis omnem aeqvationem secvndi gradvs classificat.

Tota catena sic svmmatim exponi potest: chorda dvobvs modis compvtata cosinvm differentiae dat; ex eo omnes formvlae additionis et svbtractionis orivntvr; ex his transformationes et forma harmonica; omnibvs his aeqvationes solvvntvr; et aeqvationes solvtae angvlos rectarvm metivntvr et terminvm mixtvm sectionvm conicarvm eliminant. Pictvra qvae seqvitvr hanc catenam in qvinqve segmenta colligit, qvae ordinem paragraphorvm svperiorvm seqvvntvr: qvisqve nodvs colvmnae est definitio avt theorema, secvndvm ordinem logicvm nvmeratvs et arcv cvm anvlis antecedentibvs conivnctvs qvibvs eivs demonstratio vtitvr; circvm flvitant fvlcra, id est principia et instrvmenta ad qvae catena itervm atqve itervm recvrrit, qvodqve linea continva cvm anvlis qvi ei nitvntvr conivnctvm et linea pvnctata cvm anvlo in qvo stabilitvr, exceptis iis qvae sine demonstratione admittvntvr avt a parte antecedenti vel a cvrsv I provenivnt, qvae vt circvli vacvi pingvntvr. Fvlcro pvlsato omnes anvli qvi eo vtvntvr illvstrantvr, et qvisqve anvlvs ad lectionem remittit in qva demonstratvr.

Fvlcra catenae

  • ACircvmferentia vnitaria. Admittitvr axioma percvrsvs per longitvdinem arcvs: cviqve nvmero reali ss vnvm tantvm pvnctvm P(s)P(s) circvmferentiae vnitariae respondet ad qvod ab (1,0)(1, 0) pervenitvr longitvdine ∣s∣\abs{s} percvrsa, percvrsvs conivngvntvr, P(s+2π)=P(s)P(s + 2\pi) = P(s), abscissa ipsivs P(s)P(s) in [0,π]\icc{0}{\pi} stricte decrescit et in [π,2π]\icc{\pi}{2\pi} crescit, et omne pvnctvm circvmferentiae est aliqvod P(s)P(s). Ex hoc axiomate in parte de fvnctionibvs trigonometricis et de coordinatis polaribvs demonstratvm est omne pvnctvm circvmferentiae esse P(θ)P(\theta) pro vno tantvm θ\theta in [0,2π)\ico{0}{2\pi}, cosinvm et sinvm, coordinatas ipsivs P(θ)P(\theta), periodicos esse et paritatem et cofvnctionem implere, et in intervallis principalibvs stricte monotonos esse. In hac vnitate sine demonstratione admittitvr.
  • BFvnctiones inversae. Arcvs sinvs, arcvs cosinvs et arcvs tangens in intervallis principalibvs vnvm angvlvm dant qvi datvm sinvm, cosinvm avt tangentem habet; demonstratvm in parte de fvnctionibvs trigonometricis et de coordinatis polaribvs. In hac vnitate sine demonstratione admittitvr.
  • CCoordinatae polares. Pvnctvm coordinatarvm polarivm (r,φ)(r, \varphi) coordinatas Cartesianas x=rcos⁡φx = r \cos \varphi et y=rsin⁡φy = r \sin \varphi habet, et distantia inter (r1,θ1)(r_1, \theta_1) et (r2,θ2)(r_2, \theta_2) implet d2=r12+r22−2r1r2cos⁡(θ1−θ2)d^2 = r_1^2 + r_2^2 - 2 r_1 r_2 \cos(\theta_1 - \theta_2); demonstratvm in parte de fvnctionibvs trigonometricis et de coordinatis polaribvs. In hac vnitate sine demonstratione admittitvr.
  • DGeometria analytica plani. Formvla distantiae, distantia pvncti a recta, theorema semiplanorvm, prodvctvm scalare cvm inaeqvalitate Cavchy-Schwarziana et angvlvs dvorvm vectorvm, et classificatio sectionvm conicarvm qvarvm axes axibvs coordinatis paralleli svnt; demonstrata in cvrsv I, in parte de geometria analytica plani, et in lectione de solvtione triangvlorvm. In hac vnitate sine demonstratione admittitvr.
  • EAlgebra et arithmetica antecedentes. Factorizatio polynomiorvm, theorema factoris et divisio synthetica, tabvlae signorvm, svmmae telescopicae et lemma Evclidis; demonstrata in cvrsv I, in partibvs de nvmeris natvralibvs et de fvnctionibvs algebraicis et polynomiis. In hac vnitate sine demonstratione admittitvr.
  • FIdentitas Pythagorica. Pro omni θ\theta valet sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1, et in eorvm dominiis 1+tan⁡2θ=sec⁡2θ1 + \tan^2 \theta = \sec^2 \theta et 1+cot⁡2θ=csc⁡2θ1 + \cot^2 \theta = \csc^2 \theta. Constitvitvr in: 2. Identitates fvndamentales.
  • GFormvlae additionis et svbtractionis. Pro omnibvs realibvs α\alpha et β\beta valent cos⁡(α+β)=cos⁡αcos⁡β−sin⁡αsin⁡β\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta et sin⁡(α+β)=sin⁡αcos⁡β+cos⁡αsin⁡β\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta, atqve formvlae differentiae, qvae orivntvr cvm β\beta in locvm ipsivs −β-\beta ponitvr. Constitvitvr in: 6. Formvlae additionis et svbtractionis.
  • HGeometria Evclidea. Ex geometria Evclidea admittvntvr, vt in parte cvrsvs I de geometria analytica plani, theorema Pythagorae eivsqve conversvm, aeqvalitas angvlorvm correspondentivm qvos secans cvm dvabvs parallelis facit, et svmma π\pi angvlorvm triangvli; ex eis dedvcvntvr aeqvalitas angvlorvm ad verticem oppositorvm et angvlorvm alternorvm inter parallelas. In hac vnitate sine demonstratione admittitvr.

  1. Segmentvm 1 Lingva: identitates earvmqve demonstratio

    • Identitatem definire vt aeqvalitatem qvae in toto dominio commvni membrorvm valet, eamqve ab aeqvatione distingvere.
    • Identitates reciprocas, qvotientis, Pythagoricas, paritatis et cofvnctionis colligere.
    • Identitates demonstrare non ab eo proficiscendo qvod demonstrandvm est, et falsas exemplo contrario refvtare.
    • Radices per svbstitvtiones transformare et ellipsim atqve hyperbolam in earvm parametrizationibvs trigonometricis agnoscere.

    1. Identitas eivsqve demonstratio. Identitas est aeqvalitas qvae in toto dominio commvni dvorvm membrorvm valet; demonstratvr altervm membrvm in altervm transformando, avt vtrvmqve in eandem expressionem, non ab ipsa proficiscendo, et vno exemplo contrario refvtatvr.

      Demonstratvr in: Identitates trigonometricae fvndamentales.

    2. Identitates fvndamentales. In dominio commvni cvivsqve valent identitates reciprocae, qvotientis et Pythagoricae; sinvs, tangens, cosecans et cotangens impares svnt, cosinvs et secans pares, et qvaeqve fvnctio ipsivs π2−θ\frac{\pi}{2} - \theta est eivs cofvnctio ipsivs θ\theta.

      Vtitvr: 1. Identitas eivsqve demonstratio. Nititvr: A (Circvmferentia vnitaria). Demonstratvr in: Identitates trigonometricae fvndamentales.

    3. Svbstitvtiones trigonometricae. Si θ=arcsin⁡x\theta = \arcsin x, tvm 1−x2=cos⁡θ\sqrt{1 - x^2} = \cos \theta; si θ=arctan⁡x\theta = \arctan x, tvm 1+x2=sec⁡θ\sqrt{1 + x^2} = \sec \theta; et si x≥1x \geq 1 et θ=arccos⁡1x\theta = \arccos \frac{1}{x}, tvm x2−1=tan⁡θ\sqrt{x^2 - 1} = \tan \theta.

      Vtitvr: 2. Identitates fvndamentales. Nititvr: F (Identitas Pythagorica), B (Fvnctiones inversae). Demonstratvr in: Identitates trigonometricae fvndamentales.

    4. Ellipsis et hyperbola parametricae. Dvm tt intervallvm [0,2π)\ico{0}{2\pi} percvrrit, pvnctvm (acos⁡t,bsin⁡t)(a \cos t, b \sin t) ellipsim x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 semel percvrrit; et pvnctvm (asec⁡t,btan⁡t)(a \sec t, b \tan t) ramvm dextrvm hyperbolae x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 semel percvrrit dvm tt intervallvm (−π2,π2)\ioo{-\frac{\pi}{2}}{\frac{\pi}{2}} percvrrit, et sinistrvm dvm (π2,3π2)\ioo{\frac{\pi}{2}}{\frac{3\pi}{2}} percvrrit.

      Vtitvr: 2. Identitates fvndamentales. Nititvr: A (Circvmferentia vnitaria), F (Identitas Pythagorica). Demonstratvr in: Identitates trigonometricae fvndamentales.

  2. Segmentvm 2 Formvla cosinvs differentiae eivsqve conseqventia

    • Cosinvm differentiae demonstrare chordam circvmferentiae vnitariae dvobvs modis compvtando.
    • Ex eo cosinvm et sinvm svmmae et differentiae, et tangentem svmmae dedvcere.
    • Formvlas angvli dvpli, redvctionis potentiarvm, angvli dimidii et angvli triplicis obtinere.
    • Sinvm et cosinvm vt fractiones rationales ipsivs t=tan⁡θ2t = \tan \frac{\theta}{2} exprimere, circvmferentiam rationaliter parametrizare et omnes ternas Pythagoricas generare.

    1. Cosinvs differentiae. Pro omnibvs 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, qvia chorda qvae P(α)P(\alpha) cvm P(β)P(\beta) conivngit eandem longitvdinem habet in coordinatis Cartesianis et in coordinatis polaribvs compvtata.

      Nititvr: C (Coordinatae polares), D (Geometria analytica plani), F (Identitas Pythagorica). Demonstratvr in: Formvlae additionis et svbtractionis, angvli dvpli et dimidii.

    2. Formvlae additionis et svbtractionis. Paritate et cofvnctione obtinentvr cos⁡(α+β)=cos⁡αcos⁡β−sin⁡αsin⁡β\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta, sin⁡(α+β)=sin⁡αcos⁡β+cos⁡αsin⁡β\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta et sinvs differentiae; et, si tangentes ipsorvm α\alpha et β\beta definitae svnt et tan⁡αtan⁡β≠1\tan \alpha \tan \beta \neq 1, tvm tan⁡(α+β)=tan⁡α+tan⁡β1−tan⁡αtan⁡β\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}.

      Vtitvr: 2. Identitates fvndamentales, 5. Cosinvs differentiae. Demonstratvr in: Formvlae additionis et svbtractionis, angvli dvpli et dimidii.

    3. Angvlvs dvplex. Valent sin⁡2θ=2sin⁡θcos⁡θ\sin 2\theta = 2 \sin \theta \cos \theta, cos⁡2θ=cos⁡2θ−sin⁡2θ=2cos⁡2θ−1=1−2sin⁡2θ\cos 2\theta = \cos^2 \theta - \sin^2 \theta = 2 \cos^2 \theta - 1 = 1 - 2 \sin^2 \theta et, vbi definita est, tan⁡2θ=2tan⁡θ1−tan⁡2θ\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta}.

      Nititvr: G (Formvlae additionis et svbtractionis), F (Identitas Pythagorica). Demonstratvr in: Formvlae additionis et svbtractionis, angvli dvpli et dimidii.

    4. Redvctio potentiarvm et angvlvs dimidivs. Valent cos⁡2θ=1+cos⁡2θ2\cos^2 \theta = \frac{1 + \cos 2\theta}{2} et sin⁡2θ=1−cos⁡2θ2\sin^2 \theta = \frac{1 - \cos 2\theta}{2}; ex eis sinvs et cosinvs ipsivs θ2\frac{\theta}{2} svnt radices ipsarvm 1−cos⁡θ2\frac{1 - \cos \theta}{2} et 1+cos⁡θ2\frac{1 + \cos \theta}{2} cvm signo qvadrantis svi, et tan⁡θ2=sin⁡θ1+cos⁡θ\tan \frac{\theta}{2} = \frac{\sin \theta}{1 + \cos \theta} si cos⁡θ≠−1\cos \theta \neq -1.

      Vtitvr: 7. Angvlvs dvplex. Nititvr: F (Identitas Pythagorica). Demonstratvr in: Formvlae additionis et svbtractionis, angvli dvpli et dimidii.

    5. Angvlvs triplex. Pro omni reali θ\theta valent cos⁡3θ=4cos⁡3θ−3cos⁡θ\cos 3\theta = 4 \cos^3 \theta - 3 \cos \theta et sin⁡3θ=3sin⁡θ−4sin⁡3θ\sin 3\theta = 3 \sin \theta - 4 \sin^3 \theta.

      Vtitvr: 7. Angvlvs dvplex. Nititvr: F (Identitas Pythagorica), G (Formvlae additionis et svbtractionis). Demonstratvr in: Formvlae additionis et svbtractionis, angvli dvpli et dimidii.

    6. Tangens angvli dimidii. Si θ\theta non est mvltiplvm impar ipsivs π\pi et t=tan⁡θ2t = \tan \frac{\theta}{2}, tvm sin⁡θ=2t1+t2\sin \theta = \frac{2t}{1 + t^2} et cos⁡θ=1−t21+t2\cos \theta = \frac{1 - t^2}{1 + t^2}.

      Vtitvr: 7. Angvlvs dvplex. Nititvr: F (Identitas Pythagorica). Demonstratvr in: Formvlae additionis et svbtractionis, angvli dvpli et dimidii.

    7. Parametrizatio rationalis. Recta declivitatis tt per (−1,0)(-1, 0) transiens circvmferentiam vnitariam itervm secat in (1−t21+t2,2t1+t2)\left( \frac{1 - t^2}{1 + t^2}, \frac{2t}{1 + t^2} \right); omne alivd pvnctvm circvmferentiae sic obtinetvr pro vno tantvm tt, et rationale est si et solvm si tt rationale est.

      Vtitvr: 10. Tangens angvli dimidii. Nititvr: E (Algebra et arithmetica antecedentes). Demonstratvr in: Formvlae additionis et svbtractionis, angvli dvpli et dimidii.

    8. Ternae Pythagoricae. Ternae Pythagoricae primitivae cvm bb pari svnt exacte a=m2−n2a = m^2 - n^2, b=2mnb = 2mn et c=m2+n2c = m^2 + n^2, cvm m>nm > n, mcd⁡(m,n)=1\operatorname{mcd}(m, n) = 1 et m−nm - n impari.

      Vtitvr: 11. Parametrizatio rationalis. Nititvr: E (Algebra et arithmetica antecedentes). Demonstratvr in: Formvlae additionis et svbtractionis, angvli dvpli et dimidii.

  3. Segmentvm 3 Transformationes et forma harmonica

    • Prodvcta in svmmas et svmmas in prodvcta convertere, formvlas additionis et svbtractionis addendo et svbtrahendo.
    • Omnem combinationem asin⁡x+bcos⁡xa \sin x + b \cos x vt vnam vndam Rsin⁡(x+φ)R \sin(x + \varphi) scribere eivsqve copiam imaginvm obtinere.
    • Oscillationes eivsdem freqventiae componere et pvlsationes explicare.
    • Sinvs et cosinvs in progressione arithmetica per svmmam telescopicam addere.

    1. De prodvcto in svmmam. Valet sin⁡αcos⁡β=12(sin⁡(α+β)+sin⁡(α−β))\sin \alpha \cos \beta = \frac{1}{2} \left( \sin(\alpha + \beta) + \sin(\alpha - \beta) \right), et eodem modo cos⁡αsin⁡β\cos \alpha \sin \beta, cos⁡αcos⁡β\cos \alpha \cos \beta et sin⁡αsin⁡β\sin \alpha \sin \beta svnt semisvmmae avt semidifferentiae sinvvm avt cosinvvm ipsorvm α+β\alpha + \beta et α−β\alpha - \beta.

      Nititvr: G (Formvlae additionis et svbtractionis). Demonstratvr in: Transformatio svmmarvm in prodvcta et forma harmonica.

    2. De svmma in prodvctvm. Pro omnibvs realibvs pp et qq valet sin⁡p+sin⁡q=2sin⁡p+q2cos⁡p−q2\sin p + \sin q = 2 \sin \frac{p + q}{2} \cos \frac{p - q}{2}, cvm formvlis similibvs pro sin⁡p−sin⁡q\sin p - \sin q, cos⁡p+cos⁡q\cos p + \cos q et cos⁡p−cos⁡q\cos p - \cos q; inde seqvvntvr pvlsationes ipsivs sin⁡(ax)+sin⁡(bx)\sin(ax) + \sin(bx).

      Vtitvr: 13. De prodvcto in svmmam. Demonstratvr in: Transformatio svmmarvm in prodvcta et forma harmonica.

    3. Forma harmonica. Si aa et bb non ambo nvlla svnt et R=a2+b2R = \sqrt{a^2 + b^2}, exstat vnvm tantvm φ\varphi in [0,2π)\ico{0}{2\pi} ita vt asin⁡x+bcos⁡x=Rsin⁡(x+φ)a \sin x + b \cos x = R \sin(x + \varphi) pro omni xx; haec forma vnica est, et fvnctio copiam imaginvm [−R,R]\icc{-R}{R} habet.

      Nititvr: A (Circvmferentia vnitaria), D (Geometria analytica plani), F (Identitas Pythagorica), G (Formvlae additionis et svbtractionis). Demonstratvr in: Transformatio svmmarvm in prodvcta et forma harmonica.

    4. Svperpositio oscillationvm. Nisi se invicem tollant, oscillationes A1sin⁡(ωt+φ1)A_1 \sin(\omega t + \varphi_1) et A2sin⁡(ωt+φ2)A_2 \sin(\omega t + \varphi_2) svmmam habent oscillationem eivsdem freqventiae, cvivs amplitvdo implet A2=A12+A22+2A1A2cos⁡(φ1−φ2)A^2 = A_1^2 + A_2^2 + 2 A_1 A_2 \cos(\varphi_1 - \varphi_2).

      Vtitvr: 5. Cosinvs differentiae, 15. Forma harmonica. Nititvr: F (Identitas Pythagorica), G (Formvlae additionis et svbtractionis). Demonstratvr in: Transformatio svmmarvm in prodvcta et forma harmonica.

    5. Svmmae in progressione arithmetica. Si xx non est mvltiplvm integrvm ipsivs 2π2\pi, valet ∑k=1nsin⁡(kx)=sin⁡nx2sin⁡(n+1)x2sin⁡x2\sum_{k=1}^{n} \sin(kx) = \frac{\sin \frac{nx}{2} \sin \frac{(n + 1)x}{2}}{\sin \frac{x}{2}}, et svmma cosinvvm obtinetvr cvm cos⁡(n+1)x2\cos \frac{(n + 1)x}{2} loco ipsivs sin⁡(n+1)x2\sin \frac{(n + 1)x}{2}.

      Vtitvr: 13. De prodvcto in svmmam, 14. De svmma in prodvctvm. Nititvr: E (Algebra et arithmetica antecedentes). Demonstratvr in: Transformatio svmmarvm in prodvcta et forma harmonica.

  4. Segmentvm 4 Aeqvationes et inaeqvationes

    • Solvtiones generales ipsarvm sin⁡x=c\sin x = c, cos⁡x=c\cos x = c et tan⁡x=c\tan x = c ex fvnctionibvs inversis et periodicitate obtinere.
    • Ceteras aeqvationes ad elementares redigere per identitates, factorizationem, aeqvalitatem dvorvm sinvvm avt dvorvm cosinvvm et formam harmonicam.
    • Nec solvtiones dividendo amittere nec qvadrando addere.
    • Inaeqvationes in vna conversione solvere, circvmferentia vnitaria avt tabvlis signorvm, easqve per periodicitatem extendere.

    1. Solvtiones generales. Si ∣c∣≤1\abs{c} \leq 1, solvtiones ipsivs sin⁡x=c\sin x = c svnt arcsin⁡c+2kπ\arcsin c + 2k\pi et π−arcsin⁡c+2kπ\pi - \arcsin c + 2k\pi, et ipsivs cos⁡x=c\cos x = c svnt arccos⁡c+2kπ\arccos c + 2k\pi et −arccos⁡c+2kπ-\arccos c + 2k\pi; pro omni reali cc, solvtiones ipsivs tan⁡x=c\tan x = c svnt arctan⁡c+kπ\arctan c + k\pi, cvm kk integro.

      Vtitvr: 2. Identitates fvndamentales. Nititvr: A (Circvmferentia vnitaria), B (Fvnctiones inversae). Demonstratvr in: Aeqvationes et inaeqvationes trigonometricae.

    2. Aeqvalitas sinvvm et cosinvvm. Valet sin⁡A=sin⁡B\sin A = \sin B si et solvm si A=B+2kπA = B + 2k\pi avt A=π−B+2kπA = \pi - B + 2k\pi, et cos⁡A=cos⁡B\cos A = \cos B si et solvm si A=B+2kπA = B + 2k\pi avt A=−B+2kπA = -B + 2k\pi, pro aliqvo integro kk; dvae tangentes definitae aeqvales svnt si et solvm si A=B+kπA = B + k\pi.

      Vtitvr: 14. De svmma in prodvctvm. Nititvr: A (Circvmferentia vnitaria), G (Formvlae additionis et svbtractionis). Demonstratvr in: Aeqvationes et inaeqvationes trigonometricae.

    3. Aeqvationes polynomiales et homogeneae. Aeqvatio P(sin⁡x)=0P(\sin x) = 0 aeqvivalet aeqvationibvs sin⁡x=r\sin x = r pro radicibvs realibvs rr ipsivs PP; aeqvatio homogenea in sin⁡x\sin x et cos⁡x\cos x ad polynomialem in tan⁡x\tan x redigitvr zeris cosinvs seorsvm examinatis; et omnis solvtio qvadrando obtenta confirmanda est.

      Vtitvr: 18. Solvtiones generales. Nititvr: E (Algebra et arithmetica antecedentes). Demonstratvr in: Aeqvationes et inaeqvationes trigonometricae.

    4. Aeqvatio linearis in sinv et cosinv. Aeqvatio asin⁡x+bcos⁡x=ca \sin x + b \cos x = c solvtionem habet si et solvm si c2≤a2+b2c^2 \leq a^2 + b^2, et tvm eivs solvtiones svnt −φ+arcsin⁡cR+2kπ-\varphi + \arcsin \frac{c}{R} + 2k\pi et π−φ−arcsin⁡cR+2kπ\pi - \varphi - \arcsin \frac{c}{R} + 2k\pi, vbi RR et φ\varphi svnt ea formae harmonicae.

      Vtitvr: 15. Forma harmonica, 18. Solvtiones generales. Demonstratvr in: Aeqvationes et inaeqvationes trigonometricae.

    5. Inaeqvationes. Si −1<c<1-1 < c < 1 et β=arccos⁡c\beta = \arccos c, valet cos⁡x>c\cos x > c si et solvm si −β+2kπ<x<β+2kπ-\beta + 2k\pi < x < \beta + 2k\pi pro aliqvo integro kk, et simile pro sinv valet; prodvcta et qvotientes factorvm trigonometricorvm tabvla signorvm in vna conversione solvvntvr.

      Vtitvr: 18. Solvtiones generales. Nititvr: A (Circvmferentia vnitaria), B (Fvnctiones inversae), E (Algebra et arithmetica antecedentes). Demonstratvr in: Aeqvationes et inaeqvationes trigonometricae.

  5. Segmentvm 5 Geometria analytica: rectae et sectiones conicae

    • Angvlvm dvarvm rectarvm tangente differentiae et prodvcto scalari metiri.
    • Bisectrices dvarvm rectarvm et incentrvm triangvli obtinere.
    • Altitvdines inaccessas, motvm harmonicvm, spativm iactvs proiectilis et reflexionem lvcis ad formam mathematicam redigere.
    • Formvlas rotationis dedvcere, terminvm in xyxy eliminare et omnem aeqvationem secvndi gradvs per signvm discriminantis classificare.

    1. Angvlvs dvarvm rectarvm. Declivitas rectae non verticalis est tangens eivs inclinationis; dvae rectae declivitatvm m1m_1 et m2m_2 perpendicvlares svnt si et solvm si 1+m1m2=01 + m_1 m_2 = 0, et si non svnt, angvlvs θ\theta inter eas implet tan⁡θ=∣m2−m11+m1m2∣\tan \theta = \abs{\frac{m_2 - m_1}{1 + m_1 m_2}}, congrventer cvm formvla prodvcti scalaris vectorvm directionis earvm.

      Vtitvr: 5. Cosinvs differentiae, 6. Formvlae additionis et svbtractionis. Nititvr: B (Fvnctiones inversae), D (Geometria analytica plani), F (Identitas Pythagorica), H (Geometria Evclidea). Demonstratvr in: Angvlvs dvarvm rectarvm et vsvs trigonometriae.

    2. Bisectrices et incentrvm. Pvncta qvae a dvabvs rectis se secantibvs aeqve distant dvas rectas inter se perpendicvlares formant, bisectrices, qvarvm qvaeqve alivd par angvlorvm ad verticem oppositorvm bifariam dividit; et tres bisectrices interiores triangvli, qvarvm qvaeqve latvs oppositvm secat, in pvncto interiore concvrrvnt, incentro, centro circvmferentiae inscriptae.

      Vtitvr: 8. Redvctio potentiarvm et angvlvs dimidivs. Nititvr: D (Geometria analytica plani), H (Geometria Evclidea). Demonstratvr in: Angvlvs dvarvm rectarvm et vsvs trigonometriae.

    3. Spativm iactvs proiectilis. Admissis ex physica aeqvationibvs motvs, qvae progressvm horizontalem aeqvabilem cvm casv aeqvabiliter accelerato conivngvnt, spativm iactvs proiectilis celeritate vv et elevatione θ\theta emissi est v2sin⁡2θg\frac{v^2 \sin 2\theta}{g}; maximvm est pro θ=π4\theta = \frac{\pi}{4}, et qvodqve spativm minvs dvobvs exacte angvlis obtinetvr, qvi complementarii svnt.

      Vtitvr: 18. Solvtiones generales, 7. Angvlvs dvplex. Demonstratvr in: Angvlvs dvarvm rectarvm et vsvs trigonometriae.

    4. Formvlae rotationis. Si axes angvlo θ\theta rotantvr, coordinatae novae svnt X=xcos⁡θ+ysin⁡θX = x \cos \theta + y \sin \theta et Y=−xsin⁡θ+ycos⁡θY = -x \sin \theta + y \cos \theta, et distantia dvorvm pvnctorvm non mvtatvr.

      Nititvr: A (Circvmferentia vnitaria), C (Coordinatae polares), F (Identitas Pythagorica), G (Formvlae additionis et svbtractionis). Demonstratvr in: Rotatio axivm et aeqvatio generalis secvndi gradvs.

    5. Coefficientes post rotationem. Axibvs rotatis aeqvatio Ax2+Bxy+Cy2+Dx+Ey+F=0Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 terminvm constantem servat, et eivs novvs coefficiens ipsivs XYXY est B1=Bcos⁡2θ−(A−C)sin⁡2θB_1 = B \cos 2\theta - (A - C) \sin 2\theta.

      Vtitvr: 26. Formvlae rotationis, 7. Angvlvs dvplex. Demonstratvr in: Rotatio axivm et aeqvatio generalis secvndi gradvs.

    6. Eliminatio termini in xy. Si B≠0B \neq 0, exstat vnvs tantvm angvlvs θ\theta in (0,π2)\ioo{0}{\frac{\pi}{2}} cvivs rotatio B1B_1 evanescere facit: is qvi cot⁡2θ=A−CB\cot 2\theta = \frac{A - C}{B} implet.

      Vtitvr: 27. Coefficientes post rotationem. Nititvr: B (Fvnctiones inversae). Demonstratvr in: Rotatio axivm et aeqvatio generalis secvndi gradvs.

    7. Invariantia rotationis. Pro omni angvlo rotationis valent A1+C1=A+CA_1 + C_1 = A + C et B12−4A1C1=B2−4ACB_1^2 - 4A_1C_1 = B^2 - 4AC.

      Vtitvr: 27. Coefficientes post rotationem, 7. Angvlvs dvplex. Nititvr: F (Identitas Pythagorica). Demonstratvr in: Rotatio axivm et aeqvatio generalis secvndi gradvs.

    8. Classificatio per discriminantem. Si AA, BB et CC non omnes nvlli svnt et Δ=B2−4AC\Delta = B^2 - 4AC, aeqvatio ellipsim, pvnctvm avt nvllvm pvnctvm repraesentat si Δ<0\Delta < 0; parabolam, dvas rectas parallelas, vnam rectam avt nvllvm pvnctvm si Δ=0\Delta = 0; et hyperbolam avt dvas rectas se secantes si Δ>0\Delta > 0.

      Vtitvr: 26. Formvlae rotationis, 28. Eliminatio termini in xy, 29. Invariantia rotationis. Nititvr: D (Geometria analytica plani). Demonstratvr in: Rotatio axivm et aeqvatio generalis secvndi gradvs.

Ad nvmeros complexos forma polari. Cvm cos⁡α+isin⁡α\cos \alpha + i \sin \alpha per cos⁡β+isin⁡β\cos \beta + i \sin \beta mvltiplicatvr, cosinvs et sinvs ipsivs α+β\alpha + \beta apparent: formvlae additionis et svbtractionis svnt regvla mvltiplicationis nvmerorvm complexorvm, et pars qvae seqvitvr eas in vnam colligit, formvlam Moivreanam.