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Rationes trigonometricae et circvmferentia vnitaria

Svmmarivm

Haec lectio sinvm, cosinvm et tangentem angvli dvobvs modis definit. Primvs, vt qvotientes latervm triangvli rectangvli, ad angvlos acvtos valet, et per similitvdinem demonstratvr evm a triangvlo electo non pendere. Alter, vt coordinatae pvncti in qvo latvs terminale angvli circvmferentiam vnitariam secat, ad omnem nvmervm realem valet et cvm primo in angvlis acvtis congrvit. Ex eo dedvcvntvr signa in singvlis qvadrantibvs, identitas Pythagorica, valores insignes angvlorvm π6\frac{\pi}{6}, π4\frac{\pi}{4} et π3\frac{\pi}{3}, qvi ex dimidio triangvlo aeqvilatero et ex dimidio qvadrato obtinentvr, et symmetriae sinvs et cosinvs, qvae pvnctvm in circvmferentia reflectendo demonstrantvr et rationes cvivsvis angvli per angvlvm referentiae eivs compvtare sinvnt.

Proposita discendi

  1. Sex rationes trigonometricas angvli acvti in triangvlo rectangvlo definire et per similitvdinem demonstrare eas a triangvlo non pendere.
  2. Sinvm, cosinvm ceterasqve rationes angvli cvivsvis per circvmferentiam vnitariam definire, tangentem vt declivitatem lateris terminalis agnoscere et demonstrare vtramqve definitionem in angvlis acvtis congrvere.
  3. Identitatem Pythagoricam, valores insignes et symmetrias sinvs et cosinvs demonstrare, eisqve vti ad rationes cvivsvis angvli methodo angvli referentiae compvtandas.
  4. Omnes rationes angvli ex vna earvm et ex qvadrante obtinere, et rationes ad problemata altitvdinvm, distantiarvm et arearvm adhibere.

Rationes angvli acvti

In triangvlo rectangvlo latvs angvlo recto oppositvm hypotenvsa vocatvr, cetera dvo catheti. Angvli interiores triangvli simvl angvlvm dvobvs rectis aeqvalem efficivnt, qvi π\pi metitvr; hoc geometriae elementaris factvm in lectione de angvlis eorvmqve mensvra admissvm est. Si hoc ita est, dvo angvli non recti triangvli rectangvli simvl π2\frac{\pi}{2} efficivnt, et vterqve eorvm est acvtvs, id est inter 00 et π2\frac{\pi}{2} metitvr. Respectv vnivs horvm angvlorvm acvtorvm, mensvrae α\alpha, altera cathetvs est cathetvs opposita, qvae verticem eivs non tangit, altera cathetvs adiacens, qvae cvm hypotenvsa latera angvli efficit.

Sit α\alpha angvlvs acvtvs triangvli rectangvli hypotenvsae cc, cvivs cathetvs angvlo α\alpha opposita aa metitvr et cvivs cathetvs adiacens bb metitvr. Sinvs, cosinvs et tangens angvli α\alpha svnt, ordine, qvotiens catheti oppositae per hypotenvsam, catheti adiacentis per hypotenvsam, et catheti oppositae per adiacentem.
sin⁡α=ac,cos⁡α=bc,tan⁡α=ab\sin \alpha = \frac{a}{c}, \qquad \cos \alpha = \frac{b}{c}, \qquad \tan \alpha = \frac{a}{b}

Earvm reciprocae cosecans, secans et cotangens vocantvr: csc⁡α=ca\csc \alpha = \frac{c}{a}, sec⁡α=cb\sec \alpha = \frac{c}{b} et cot⁡α=ba\cot \alpha = \frac{b}{a}. Sex hae rationes trigonometricae vocantvr, qvia rationes, id est qvotientes, latervm triangvli svnt. Definitio tamen dvbitationem movet: infinita svnt triangvla rectangvla angvlvm acvtvm mensvrae α\alpha habentia, magna et parva, et si singvla qvotientem diversvm darent, verba «sinvs angvli α\alpha» nvllvm nvmervm designarent. Theorema seqvens dvbitationem tollit.

TheoremaRationes a triangvlo non pendent

Si dvo triangvla rectangvla angvlvm acvtvm eivsdem mensvrae α\alpha habent, qvotiens dvorvm qvorvmlibet latervm primi aeqvalis est qvotienti latervm respondentivm alterivs. Itaqve sinvs, cosinvs et tangens angvli α\alpha, et earvm reciprocae, ab angvlo α\alpha solo pendent. Praeterea 0<sin⁡α<10 < \sin \alpha < 1 et 0<cos⁡α<10 < \cos \alpha < 1.

Sint dvo triangvla rectangvla angvlvm acvtvm α\alpha habentia, primvm cathetis a1a_1 et b1b_1 et hypotenvsa c1c_1, altervm cathetis a2a_2 et b2b_2 et hypotenvsa c2c_2, vbi a1a_1 et a2a_2 svnt catheti angvlo α\alpha oppositae. Admittvntvr criteria similitvdinis triangvlorvm geometriae elementaris, eadem qvae in lectione cvrsvs I de aeqvatione rectae declivitatem a pvnctis electis non pendere praestitervnt.

Demonstratio

  1. a2=ka1,b2=kb1,c2=kc1\dato{k}{a_2 = k a_1, \quad b_2 = k b_1, \quad c_2 = k c_1}

    Dvo triangvla dvos angvlos aeqvales habent, rectvm et evm mensvrae α\alpha; per criterivm similitvdinis dvorvm angvlorvm similia svnt, et latera eorvm respondentia proportionalia svnt: est nvmervs k>0k > 0 talis vt qvodqve latvs alterivs sit kk vicibvs latvs respondens primi.

  2. a2c2=ka1kc1=a1c1\frac{a_2}{c_2} = \frac{\resaltar{k a_1}}{\resaltar{k c_1}} = \resaltar{\frac{a_1}{c_1}}

    Qvotiens catheti oppositae per hypotenvsam idem est in vtroqve triangvlo, qvia factor kk tollitvr.

  3. b2c2=kb1kc1=b1c1\frac{b_2}{c_2} = \frac{\resaltar{k b_1}}{\resaltar{k c_1}} = \resaltar{\frac{b_1}{c_1}}

    Idem accidit qvotienti catheti adiacentis per hypotenvsam.

  4. a2b2=ka1kb1=a1b1\frac{a_2}{b_2} = \frac{\resaltar{k a_1}}{\resaltar{k b_1}} = \resaltar{\frac{a_1}{b_1}}

    Et qvotienti dvarvm cathetorvm. Itaqve sinvs, cosinvs et tangens, et cvm eis earvm reciprocae, evndem valorem in dvobvs triangvlis capivnt: ab angvlo α\alpha solo pendent.

  5. c2=a2+b2→0<ac<1,0<bc<1c^2 = a^2 + b^2 \rightarrow 0 < \frac{a}{c} < 1, \quad 0 < \frac{b}{c} < 1

    Per theorema Pythagorae qvadratvm hypotenvsae est svmma qvadratorvm cathetorvm, et vtrvmqve eorvm svperat: hypotenvsa maior est vtraqve catheto, et dvo qvotientes per hypotenvsam inter 00 et 11 iacent.

Theorema sinit rationes angvli qvolibet triangvlo rectangvlo evm continente compvtare, et commodissimvm eligere. Praesertim cathetvs vni angvlorvm acvtorvm opposita est cathetvs alteri adiacens, et inde seqvitvr relatio inter rationes dvorvm angvlorvm qvi simvl π2\frac{\pi}{2} efficivnt, qvi complementares vocantvr.

ExemplvmRationes dvorvm angvlorvm acvtorvm

In triangvlo rectangvlo cathetis 33 et 44, sit α\alpha angvlvs catheto 33 oppositvs et β\beta angvlvs catheto 44 oppositvs. Rationes eorvm compvtare et comparare.

Demonstratio

  1. c=32+42=25=5c = \sqrt{\resaltar{3^2 + 4^2}} = \sqrt{25} = \resaltar{5}

    Per theorema Pythagorae hypotenvsa 55 metitvr.

  2. sin⁡α=35,cos⁡α=45,tan⁡α=34\dato{a}{\sin \alpha = \frac{3}{5}, \quad \cos \alpha = \frac{4}{5}, \quad \tan \alpha = \frac{3}{4}}

    Pro angvlo α\alpha, cathetvs opposita est 33 et adiacens est 44.

  3. sin⁡β=45,cos⁡β=35,tan⁡β=43\dato{b}{\sin \beta = \frac{4}{5}, \quad \cos \beta = \frac{3}{5}, \quad \tan \beta = \frac{4}{3}}

    Pro angvlo β\beta, mvnera cathetorvm permvtantvr: opposita est 44 et adiacens est 33.

  4. sin⁡β=cos⁡α,cos⁡β=sin⁡α,tan⁡β=1tan⁡α=cot⁡α\sin \beta = \cos \alpha, \quad \cos \beta = \sin \alpha, \quad \tan \beta = \frac{1}{\tan \alpha} = \cot \alpha

    Cvm α+β=π2\alpha + \beta = \frac{\pi}{2}, sinvs vtrivsqve angvli est cosinvs alterivs, et tangens vtrivsqve est cotangens alterivs. Idem accidit in omni triangvlo rectangvlo, qvia permvtatio cathetorvm a longitvdinibvs earvm non pendet.

Hinc nomina dvcvntvr: cosinvs est sinvs complementi, cotangens est tangens complementi et cosecans est secans complementi.