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Identitates trigonometricae fvndamentales

Svmmarivm

Haec lectio trigonometriam analyticam aperit, qvae relationes algebraicas inter fvnctiones trigonometricas investigat. Accvrate definit qvid sit identitas, aeqvalitas qvae in toto dominio commvni dvorvm membrorvm svorvm valet, et identitates fvndamentales colligit: reciprocas, qvotientis, Pythagoricas, paritatis et cofvnctionis, qvae praeterea vt longitvdines segmentorvm in circvmferentia vnitaria legvntvr. Methodvm docet identitatis demonstrandae, altervm membrvm in altervm avt vtrvmqve in eandem expressionem transformando, et ostendit cvr non liceat ab identitate proficisci eamqve qvadrare; vna exceptio, exemplvm contrarivm, satis est ad identitatem falsam refellendam. Qvamqve fvnctionem per aliam exprimit, cvm signo qvod qvadrans figit, expressiones simplificat dominio servato, radices algebraicas in fvnctiones trigonometricas per svbstitvtiones convertit et, parametro eliminato, in pvncto (acos⁡t,bsin⁡t)\left( a \cos t, b \sin t \right) ellipsim et in pvncto (asec⁡t,btan⁡t)\left( a \sec t, b \tan t \right) hyperbolam agnoscit.

Proposita discendi

  1. Identitatem trigonometricam accvrate definire, dominivm eivs commvne determinare eamqve ab aeqvatione distingvere, atqve identitates fvndamentales colligere et demonstrare, etiam in earvm interpretatione geometrica vt segmenta.
  2. Identitates demonstrare altervm membrvm in altervm avt vtrvmqve in expressionem commvnem transformando, ratiocinationes vitiosas agnoscere et identitatem falsam exemplo contrario refellere.
  3. Qvamqve fvnctionem trigonometricam per aliam exprimere, cvm signo qvod qvadranti respondet, et expressiones dominio earvm servato simplificare.
  4. Svbstitvtionibvs trigonometricis vti ad expressiones cvm radicibvs transformandas, et parametrvm eliminare in parametrizationibvs trigonometricis ellipsis et hyperbolae.

Qvid sit identitas

Qvaedam aeqvalitates inter fvnctiones trigonometricas pro omni nvmero reali valent, vt sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1, demonstrata in lectione de rationibvs trigonometricis et circvmferentia vnitaria; aliae, vt sin⁡θ=−12\sin \theta = -\frac{1}{2}, pro qvibvsdam tantvm valoribvs, qvos in lectione de fvnctionibvs trigonometricis inversis invenire didicimvs. Differentia natvrae est: prima aeqvalitas aliqvid de omnibvs nvmeris affirmat, altera qvaestionem ponit. Atqvi verba «pro omni nvmero reali» accvrativs definienda svnt, qvia mvltae expressiones trigonometricae non in omnibvs nvmeris definivntvr.

Identitas trigonometrica est aeqvalitas inter dvas expressiones ex fvnctionibvs trigonometricis constrvctas qvae pro omnibvs valoribvs variabilis valet in qvibvs vtrvmqve membrvm definitvm est. Copia horvm valorvm est dominivm commvne identitatis. Aeqvalitas qvae pro qvibvsdam tantvm valoribvs dominii commvnis valet est aeqvatio, et hi valores eivs solvtiones svnt.

Dominivm non res parva est. Tangens definitvr vbi cosinvs non evanescit, et cotangens vbi sinvs non evanescit; ergo prodvctvm vtrivsqve definitvm est exacte cvm xx non est mvltiplvm integrvm ipsivs π2\frac{\pi}{2}, et ibi valet 11:

tan⁡xcot⁡x=1,x≠kπ2\tan x \cot x = 1, \qquad x \neq \frac{k\pi}{2}

In mvltiplis ipsivs π2\frac{\pi}{2} identitas nihil affirmat, qvia primvm eivs membrvm nvllvm nvmervm designat. Eodem modo sin⁡xtan⁡x=cos⁡x\frac{\sin x}{\tan x} = \cos x identitas est in dominio svo commvni, qvod mvltipla ipsivs π2\frac{\pi}{2} exclvdit; tamen fvnctiones f(x)=sin⁡xtan⁡xf(x) = \frac{\sin x}{\tan x} et g(x)=cos⁡xg(x) = \cos x non svnt aeqvales, qvia secvndvm lectionem cvrsvs I de operationibvs fvnctionvm dvae fvnctiones aeqvales idem dominivm habent, et altera definitvr, exempli gratia, in x=0x = 0. Identitas igitvr dvas expressiones comparat tantvm vbi vtraqve sensvm habet.

Identitates fvndamentales

Identitates infra scriptae directo ex definitionibvs et ex theorematibvs partis svperioris obtinentvr; omnes ceterae ex eis dedvcvntvr.

TheoremaIdentitates fvndamentales

In dominio commvni cvivsqve aeqvalitatis valent identitates reciprocae, qvotientis et Pythagoricae infra scriptae. Praeterea sinvs, tangens, cosecans et cotangens fvnctiones impares svnt, cosinvs avtem et secans fvnctiones pares; et qvaeqve fvnctio ipsivs π2−θ\frac{\pi}{2} - \theta est cofvnctio sva ipsivs θ\theta, vbi sinvs et cosinvs, tangens et cotangens, secans et cosecans inter se cofvnctiones svnt.

sin⁡θcsc⁡θ=1,cos⁡θsec⁡θ=1,tan⁡θcot⁡θ=1\sin \theta \csc \theta = 1, \qquad \cos \theta \sec \theta = 1, \qquad \tan \theta \cot \theta = 1
tan⁡θ=sin⁡θcos⁡θ,cot⁡θ=cos⁡θsin⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}, \qquad \cot \theta = \frac{\cos \theta}{\sin \theta}
sin⁡2θ+cos⁡2θ=1,1+tan⁡2θ=sec⁡2θ,1+cot⁡2θ=csc⁡2θ\sin^2 \theta + \cos^2 \theta = 1, \qquad 1 + \tan^2 \theta = \sec^2 \theta, \qquad 1 + \cot^2 \theta = \csc^2 \theta
sec⁡(−θ)=sec⁡θ,csc⁡(−θ)=−csc⁡θ,cot⁡(−θ)=−cot⁡θ\sec(-\theta) = \sec \theta, \qquad \csc(-\theta) = -\csc \theta, \qquad \cot(-\theta) = -\cot \theta
sec⁡(π2−θ)=csc⁡θ,csc⁡(π2−θ)=sec⁡θ,cot⁡(π2−θ)=tan⁡θ\sec\left( \frac{\pi}{2} - \theta \right) = \csc \theta, \qquad \csc\left( \frac{\pi}{2} - \theta \right) = \sec \theta, \qquad \cot\left( \frac{\pi}{2} - \theta \right) = \tan \theta

Omnia in pvncto P(θ)=(cos⁡θ,sin⁡θ)P(\theta) = \left( \cos \theta, \sin \theta \right) lectionis de rationibvs trigonometricis legvntvr: qvattvor fvnctiones reliqvae ibi vt qvotientes coordinatarvm eivs definitae svnt, et theorema symmetriarvm sinvs et cosinvs, in ea demonstratvm, pvncta symmetrica ipsivs P(θ)P(\theta) respectv axis xx et rectae y=xy = x praebet. Ceterae identitates qvotientibvs et reciprocis svmendis obtinentvr. Figvra angvlvm acvtvm adhibet.

Demonstratio

  1. tan⁡θ=sin⁡θcos⁡θ,cot⁡θ=cos⁡θsin⁡θ,sec⁡θ=1cos⁡θ,csc⁡θ=1sin⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}, \quad \cot \theta = \frac{\cos \theta}{\sin \theta}, \quad \sec \theta = \frac{1}{\cos \theta}, \quad \csc \theta = \frac{1}{\sin \theta}

    Identitates reciprocae et qvotientis svnt definitiones tangentis, cotangentis, secantis et cosecantis: qvaeqve definitvr vbi denominator eivs non evanescit, et in hoc dominio prodvctvm fvnctionis et reciprocae eivs valet 11. In figvra sinvs et cosinvs coordinatae svnt ipsivs P(θ)P(\theta).

  2. sin⁡2θ+cos⁡2θ=1→sin⁡2θcos⁡2θ+1=1cos⁡2θ,1+cos⁡2θsin⁡2θ=1sin⁡2θ\sin^2 \theta + \cos^2 \theta = 1 \rightarrow \frac{\resaltar{\sin^2 \theta}}{\cos^2 \theta} + 1 = \frac{1}{\resaltar{\cos^2 \theta}}, \quad 1 + \frac{\resaltar{\cos^2 \theta}}{\sin^2 \theta} = \frac{1}{\resaltar{\sin^2 \theta}}

    Pvnctvm P(θ)P(\theta) distat 11 ab origine: haec est prima identitas Pythagorica, in lectione de rationibvs trigonometricis demonstrata. Vbi cos⁡θ≠0\cos \theta \neq 0, vtrvmqve eivs membrvm per cos⁡2θ\cos^2 \theta dividitvr, et vbi sin⁡θ≠0\sin \theta \neq 0, per sin⁡2θ\sin^2 \theta; per identitates gradvs svperioris ceterae dvae obtinentvr.

  3. P(−θ)=(cos⁡θ,−sin⁡θ)\dato{i}{P(-\theta) = \left( \cos \theta, \resaltar{-\sin \theta} \right)}

    Pvnctvm symmetricvm ipsivs P(θ)P(\theta) respectv axis xx est P(−θ)P(-\theta): per theorema symmetriarvm cosinvs par est et sinvs impar. Praesertim −θ-\theta in dominio cvivsqve fvnctionis est exacte cvm θ\theta in eo est, qvia pvncta in qvibvs sinvs et cosinvs evanescvnt symmetrica svnt.

  4. sec⁡(−θ)=1cos⁡θ=sec⁡θ,csc⁡(−θ)=1−sin⁡θ=−csc⁡θ,cot⁡(−θ)=cos⁡θ−sin⁡θ=−cot⁡θ\sec(-\theta) = \frac{1}{\cos \theta} = \sec \theta, \quad \csc(-\theta) = \frac{1}{\resaltar{-\sin \theta}} = -\csc \theta, \quad \cot(-\theta) = \frac{\cos \theta}{\resaltar{-\sin \theta}} = -\cot \theta

    Reciprocis et qvotientibvs svmendis signvm minvs apparet in fvnctionibvs qvarvm expressio sinvm semel tantvm continet: secans par est, cosecans avtem, tangens et cotangens impares svnt.

  5. P(π2−θ)=(sin⁡θ,cos⁡θ)\dato{c}{P\left( \frac{\pi}{2} - \theta \right) = \left( \resaltar{\sin \theta}, \resaltar{\cos \theta} \right)}

    Pvnctvm symmetricvm ipsivs P(θ)P(\theta) respectv rectae y=xy = x est P(π2−θ)P\left( \frac{\pi}{2} - \theta \right): per theorema symmetriarvm haec reflexio sinvm et cosinvm permvtat.

  6. sec⁡(π2−θ)=1sin⁡θ=csc⁡θ,csc⁡(π2−θ)=1cos⁡θ=sec⁡θ,cot⁡(π2−θ)=sin⁡θcos⁡θ=tan⁡θ\sec\left( \frac{\pi}{2} - \theta \right) = \frac{1}{\resaltar{\sin \theta}} = \csc \theta, \quad \csc\left( \frac{\pi}{2} - \theta \right) = \frac{1}{\resaltar{\cos \theta}} = \sec \theta, \quad \cot\left( \frac{\pi}{2} - \theta \right) = \frac{\resaltar{\sin \theta}}{\resaltar{\cos \theta}} = \tan \theta

    Reciprocis et qvotientibvs svmendis etiam secans et cosecans, tangens et cotangens permvtantvr. Dominia inter se respondent: π2−θ\frac{\pi}{2} - \theta in dominio secantis est si et solvm si θ\theta in dominio cosecantis est, qvia cos⁡(π2−θ)=sin⁡θ\cos\left( \frac{\pi}{2} - \theta \right) = \sin \theta, et idem in ceteris accidit.

Sex fvnctiones vt segmenta

Nomina tangentis et secantis e geometria venivnt: pro angvlo acvto svnt longitvdines segmentorvm in linea tangente et in linea secante circvmferentiae vnitariae. Haec figvra classica vno in schemate sex fvnctiones et tres identitates Pythagoricas conivngit.

TheoremaSex rationes vt segmenta

Sint 0<θ<π20 < \theta < \frac{\pi}{2}, PP pvnctvm P(θ)P(\theta) circvmferentiae vnitariae, QQ pes perpendicvlaris ab eo ad axem xx dvctae, A=(1,0)A = (1, 0) et B=(0,1)B = (0, 1). Recta qvae latvs terminale continet tangentem verticalem in AA secat in pvncto TT, et tangentem horizontalem in BB in pvncto SS. Tvm cosinvs, sinvs, tangens, secans, cotangens et cosecans ipsivs θ\theta svnt, ordine, longitvdines segmentorvm qvae a OO ad QQ, a QQ ad PP, a AA ad TT, a OO ad TT, a BB ad SS et a OO ad SS pertinent; et theorema Pythagorae in triangvlo verticvm OO, AA et TT, et in triangvlo verticvm OO, BB et SS, est, ordine, secvnda et tertia identitas Pythagorica.

Figvra segmentvm post segmentvm percvrritvr. Instrvmentvm est id qvod in lectione de rationibvs trigonometricis significationem tangentis dedit: recta qvae latvs terminale continet per originem transit et declivitatem tan⁡θ\tan \theta habet, ita vt eivs aeqvatio sit y=xtan⁡θy = x \tan \theta.

Demonstratio

  1. P(θ)=(cos⁡θ,sin⁡θ),Q=(cos⁡θ,0)P(\theta) = \left( \cos \theta, \sin \theta \right), \quad Q = \left( \cos \theta, 0 \right)

    Ex definitione coordinatae ipsivs PP svnt cosinvs et sinvs; segmentvm horizontale a OO ad QQ abscissam metitvr, verticale avtem a QQ ad PP ordinatam, ambae in primo qvadrante positivae.

  2. T=(1,tan⁡θ)\dato{t}{T = \left( 1, \tan \theta \right)}

    Tangens circvmferentiae in AA est recta x=1x = 1. Svbstitvendo x=1x = 1 in aeqvationem lateris terminalis obtinetvr TT: segmentvm a AA ad TT, in linea tangente sitvm, metitvr tan⁡θ\tan \theta.

  3. T=sec⁡θ⋅(cos⁡θ,sin⁡θ)T = \resaltar{\sec \theta} \cdot \left( \cos \theta, \sin \theta \right)

    Pvnctvm TT est pvnctvm P(θ)P(\theta) per sec⁡θ\sec \theta mvltiplicatvm, qvia sec⁡θcos⁡θ=1\sec \theta \cos \theta = 1. Cvm P(θ)P(\theta) distet 11 ab origine, TT distat sec⁡θ\sec \theta: segmentvm a OO ad TT, in linea secante sitvm, secantem metitvr.

  4. 12+tan⁡2θ=sec⁡2θ\resaltar{1^2} + \resaltar{\tan^2 \theta} = \resaltar{\sec^2 \theta}

    Triangvlvm verticvm OO, AA et TT rectvm est in AA, cvm cathetis 11 et tan⁡θ\tan \theta et hypotenvsa sec⁡θ\sec \theta. Theorema Pythagorae in eo est secvnda identitas Pythagorica.

  5. S=(cot⁡θ,1)=csc⁡θ⋅(cos⁡θ,sin⁡θ)\dato{s}{S = \left( \cot \theta, 1 \right) = \csc \theta \cdot \left( \cos \theta, \sin \theta \right)}

    Tangens in BB est recta y=1y = 1. Svbstitvendo y=1y = 1 in y=xtan⁡θy = x \tan \theta fit x=cot⁡θx = \cot \theta; et cvm csc⁡θsin⁡θ=1\csc \theta \sin \theta = 1, pvnctvm SS est P(θ)P(\theta) per csc⁡θ\csc \theta mvltiplicatvm. Ergo segmentvm a BB ad SS cotangentem metitvr, et segmentvm a OO ad SS cosecantem.

  6. 12+cot⁡2θ=csc⁡2θ\resaltar{1^2} + \resaltar{\cot^2 \theta} = \resaltar{\csc^2 \theta}

    Triangvlvm verticvm OO, BB et SS rectvm est in BB, cvm cathetis 11 et cot⁡θ\cot \theta et hypotenvsa csc⁡θ\csc \theta: ex eo seqvitvr tertia identitas Pythagorica. Angvlvs eivs in OO metitvr π2−θ\frac{\pi}{2} - \theta, et in eo cotangens ipsivs θ\theta est cathetvs opposita: est tangens angvli complementaris.

In figvra etiam legitvr secantem et cosecantem angvli acvti maiores esse qvam 11, qvia svnt hypotenvsae triangvlorvm qvorvm cathetvs longitvdinem 11 habet, et tangentem minorem esse qvam secantem. Prima identitas Pythagorica est, eodem modo, theorema Pythagorae in triangvlo verticvm OO, QQ et PP.