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Solvtio triangvlorvm: theoremata sinvvm et cosinvvm

Svmmarivm

Haec lectio solvtionem triangvlorvm, qvae rationibvs trigonometricis ad triangvla rectangvla tantvm pertinebat, ad omnia triangvla extendit: datis tribvs eorvm elementis, inter latera et angvlos, reliqva determinat. Ostendit aream triangvli esse dimidivm prodvcti dvorvm latervm et sinvs angvli qvem ea efficivnt, inde theorema sinvvm eivsqve cvm radio circvmferentiae circvmscriptae nexvm dedvcit, theorema cosinvvm per coordinatas demonstrat, qvod theorema Pythagorae vt casvm particvlarem continet, et ex vtroqve formvlam Heronis obtinet, qvae aream ex tribvs lateribvs dat. His instrvmentis casvs solvtionis perscrvtatvr, et praesertim casvm ambigvvm, in qvo dvo latera et angvlvs vni eorvm oppositvs nvllvm, vnvm avt dvo triangvla determinant secvndvm discriminans et signa radicvm aeqvationis qvadraticae. Postremo angvlvm dvorvm vectorvm per prodvctvm scalare metitvr, et ostendit hanc formvlam et theorema cosinvvm dvas esse lectiones vnivs eivsdemqve rei; problemata haec effecta ad medianas, ad mensvram distantiarvm inaccessibilivm et ad diagonales parallelogrammi adhibent.

Proposita discendi

  1. Aream triangvli compvtare ex dvobvs lateribvs et angvlo qvem efficivnt, avt ex tribvs eivs lateribvs per formvlam Heronis.
  2. Theoremata sinvvm et cosinvvm demonstrare, et agnoscere vtri eorvm pro datis praesto innitendvm sit.
  3. Triangvlvm solvere datis tribvs eivs lateribvs, dvobvs lateribvs et angvlo interiecto, avt vno latere et dvobvs angvlis, et nvmervm solvtionvm discvtere cvm dantvr dvo latera et angvlvs vni eorvm oppositvs.
  4. Angvlvm dvorvm vectorvm per prodvctvm scalare compvtare, et solvtionem triangvlorvm ad problemata geometrica et mensvrae adhibere.

Notatio et res iam notae

In lectione de rationibvs trigonometricis sinvs, cosinvs et tangens angvli acvti vt qvotientes latervm triangvli rectangvli definiti svnt, ac deinde per circvmferentiam vnitariam ad omnem angvlvm extensi. Triangvlvm igitvr rectangvlvm his rationibvs et theoremate Pythagorae solvitvr. Atqvi pleraqve triangvla qvae in geometria et in mensvra agrorvm occvrrvnt nvllvm angvlvm rectvm habent; hvivs lectionis propositvm est ea omnia solvere.

In triangvlo verticvm AA, BB et CC eadem littera maivscvla et verticem et mensvram angvli interioris triangvli ad evm verticem designat, et littera minvscvla respondens longitvdinem lateris oppositi designat: aa est longitvdo lateris BCBC, bb lateris CACA et cc lateris ABAB. Area triangvli notatvr SS, et distantia inter dvo pvncta PP et QQ notatvr d⁡(P,Q)\operatorname{d}(P, Q). Vt in solvtione triangvlorvm mos est, angvli in gradibvs exprimvntvr. Atqvi fvnctiones trigonometricae pro nvmeris realibvs definitae svnt, qvi mensvrae in radianis svnt; per regvlam conversionis lectionis de angvlis eorvmqve mensvra, angvlvs gg gradvvm πg180\frac{\pi g}{180} radianos metitvr, et haec conventio adhibetvr: sin⁡g∘\sin g^\circ designat sin⁡πg180\sin \frac{\pi g}{180}, et similiter in ceteris fvnctionibvs; exempli gratia, cos⁡60∘=cos⁡π3=12\cos 60^\circ = \cos \frac{\pi}{3} = \frac{1}{2}. Eodem modo arcvs cosinvs, qvi valores in [0,π]\icc{0}{\pi} svmit, mensvram angvli in radianis reddit, qvae in gradibvs exprimitvr: aeqvalitas arccos⁡(−12)=120∘\arccos \left( -\frac{1}{2} \right) = 120^\circ significat arccos⁡(−12)=2π3\arccos \left( -\frac{1}{2} \right) = \frac{2\pi}{3}, mensvram in radianis angvli 120∘120^\circ. Hac conventione omnia qvae seqvvntvr aeqve in radianis valent, si 180∘180^\circ in π\pi mvtatvr. Admittitvr, vt in lectione de angvlis eorvmqve mensvra, res geometriae elementaris qvam hic cvrsvs non demonstrat qvia geometriam syntheticam non tractat: tres angvli triangvli dvobvs rectis aeqvales svnt.

A+B+C=180∘A + B + C = 180^\circ

Inde efficitvr qvemqve angvlvm triangvli stricte inter 0∘0^\circ et 180∘180^\circ iacere, et svmmvm vnvm eorvm rectvm avt obtvsvm esse. Ex lectionibvs hvivs partis iam traditis praeterea hae res adhibentvr:

  • Pvnctvm qvod a principio distantia rr abest, in semirecta qvae angvlvm θ\theta cvm semiaxe positivo abscissarvm efficit, coordinatas (rcos⁡θ,rsin⁡θ)\left( r \cos \theta, r \sin \theta \right) habet: est pvnctvm circvmferentiae vnitariae ipsi θ\theta respondens, per rr mvltiplicatvm.
  • Identitas Pythagorica, sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1, et symmetriae angvli svpplementarii, sin⁡(180∘−θ)=sin⁡θ\sin \left( 180^\circ - \theta \right) = \sin \theta et cos⁡(180∘−θ)=−cos⁡θ\cos \left( 180^\circ - \theta \right) = -\cos \theta.
  • Si 0∘<θ<180∘0^\circ < \theta < 180^\circ, sinvs ipsivs θ\theta positivvs est; cosinvs positivvs est si θ\theta acvtvs est, nvllvs si rectvs, negativvs si obtvsvs.
  • Cosinvs inter 0∘0^\circ et 180∘180^\circ stricte decrescit et in eo intervallo qvemqve valorem inter −1-1 et 11 semel omnino svmit; qvam ob rem, secvndvm lectionem de fvnctionibvs trigonometricis inversis, angvlvs triangvli svo cosinv determinatvr, θ=arccos⁡x\theta = \arccos x.

Sinvs contra angvlvm triangvli non determinat: angvli θ\theta et 180∘−θ180^\circ - \theta evndem sinvm habent, et fvnctio arcvs sinvs solvm acvtvm reddit. Haec inter sinvm et cosinvm dissimilitvdo in sectione paenvltima explicabit cvr qvaedam data dvo triangvla diversa admittant.