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Fvnctiones trigonometricae inversae

Svmmarivm

Haec lectio inversas sinvs, cosinvs et tangentis constrvit. Cvm hae fvnctiones periodicae sint, iniectivae non svnt, et vt inverti possint, ad intervallvm restringi debent in qvo qvemqve valorem semel tantvm capiant: sinvs ad [−π2,π2]\icc{-\frac{\pi}{2}}{\frac{\pi}{2}}, cosinvs ad [0,π]\icc{0}{\pi} et tangens ad (−π2,π2)\ioo{-\frac{\pi}{2}}{\frac{\pi}{2}}. Ex axiomate percvrsvs per longitvdinem arcvs, qvod affirmat abscissam pvncti percvrsi per primam dimidiam conversionem stricte decrescere, dedvcit cosinvm in [0,π]\icc{0}{\pi} stricte decrescentem esse; ex hoc et ex symmetriis demonstrat sinvm et tangentem in svis intervallis stricte crescentes esse, et tres restrictiones biiectivas esse; arcvm sinvs, arcvm cosinvs et arcvm tangentis definit, et eorvm delineationes vt reflexiones respectv rectae y=xy = x obtinet. Deinde identitates demonstrat qvae eos conivngvnt, diligenter distingvit sin⁡(arcsin⁡x)=x\sin(\arcsin x) = x, semper veram, ab arcsin⁡(sin⁡x)=x\arcsin(\sin x) = x, veram tantvm in intervallo principali, et aeqvationes sin⁡x=c\sin x = c, cos⁡x=c\cos x = c et tan⁡x=c\tan x = c in integra conversione solvit.

Proposita discendi

  1. Ex axiomate percvrsvs monotoniam strictam cosinvs et sinvs in eorvm intervallis principalibvs dedvcere, ea monotoniam tangentis demonstrare, et tres inversas cvm svis dominiis et copiis imaginvm definire.
  2. Delineationes arcvs sinvs, arcvs cosinvs et arcvs tangentis vt reflexiones respectv rectae y=xy = x constrvere, et eorvm valores in pvnctis insignibvs compvtare.
  3. Identitates compositionis, symmetriae et angvlorvm complementarivm fvnctionvm inversarvm demonstrare, et expressiones simplificare qvae fvnctiones trigonometricas cvm earvm inversis conivngvnt.
  4. Aeqvationes trigonometricas elementares in intervallo [0,2π)\ico{0}{2\pi} per fvnctiones inversas et per symmetrias circvmferentiae vnitariae solvere.

Restrictio fvnctionvm trigonometricarvm

Secvndvm lectionem cvrsvs I de operationibvs fvnctionvm, fvnctio inversam habet si et solvm si biiectiva est, et tvm delineatio inversae est reflexio delineationis ipsivs respectv rectae y=xy = x. Atqvi fvnctiones trigonometricae in lectione de fvnctionibvs trigonometricis earvmqve delineationibvs tractatae iniectivae non svnt: sinvs et cosinvs periodvm 2π2\pi habent, et tangens periodvm π\pi, ita vt qvisqve valor infinities iteretvr. Immo intra eandem periodvm valores iterantvr: per symmetriam angvlorvm svpplementarivm, in lectione de rationibvs trigonometricis demonstratam, valet sin⁡(π−x)=sin⁡x\sin(\pi - x) = \sin x. Qvae si ita svnt, vt hae fvnctiones invertantvr, ad intervallvm restringi debent in qvo iniectivae sint et omnes svos valores capiant. Eligvntvr intervalla qvae angvlos acvtos contineant et in qvibvs fvnctio stricte monotona sit.

TheoremaMonotonia in intervallis principalibvs

Cosinvs in [0,π]\icc{0}{\pi} stricte decrescens est; sinvs in [−π2,π2]\icc{-\frac{\pi}{2}}{\frac{\pi}{2}} stricte crescens est, et tangens in (−π2,π2)\ioo{-\frac{\pi}{2}}{\frac{\pi}{2}} stricte crescens est.

Adhibetvr definitio sinvs et cosinvs per pvnctvm P(θ)=(cos⁡θ,sin⁡θ)P(\theta) = \left( \cos \theta, \sin \theta \right) circvmferentiae vnitariae, ex lectione de rationibvs trigonometricis, et axioma percvrsvs per longitvdinem arcvs, in lectione de angvlis eorvmqve mensvra admissvm. Monotonia cosinvs est translatio proprietatis dimidiae conversionis eivs axiomatis; monotonia sinvs et tangentis ex ea demonstrantvr.

Demonstratio

  1. 0≤α<β≤π0 \leq \alpha < \beta \leq \pi

    Sint α\alpha et β\beta cvm 0≤α<β≤π0 \leq \alpha < \beta \leq \pi. Pvncta P(α)P(\alpha) et P(β)P(\beta) svnt extrema arcvvm longitvdinvm α\alpha et β\beta qvi a (1,0)(1, 0) contra motvm horologii proficiscvntvr, et ambo in prima dimidia conversione percvrsvs iacent, qvae semicircvlvm svperiorem tegit.

  2. 0≤α<β≤π→cos⁡α>cos⁡β\dato{c}{0 \leq \alpha < \beta \leq \pi \rightarrow \cos \alpha > \cos \beta}

    Per proprietatem dimidiae conversionis axiomatis, dvrante prima dimidia conversione abscissa pvncti percvrsi stricte decrescit: abscissa pvncti P(α)P(\alpha) maior est qvam abscissa pvncti P(β)P(\beta). Cvm cosinvs ex definitione sit abscissa pvncti P(θ)P(\theta), inaeqvalitas inter abscissas est inaeqvalitas inter cosinvs.

  3. sin⁡α=cos⁡(π2−α)<cos⁡(π2−β)=sin⁡β\sin \alpha = \resaltar{\cos\left( \frac{\pi}{2} - \alpha \right)} < \resaltar{\cos\left( \frac{\pi}{2} - \beta \right)} = \dato{s}{\sin \beta}

    Sint nvnc −π2≤α<β≤π2-\frac{\pi}{2} \leq \alpha < \beta \leq \frac{\pi}{2}. Eorvm complementa π2−β<π2−α\frac{\pi}{2} - \beta < \frac{\pi}{2} - \alpha satisfacivnt et in [0,π]\icc{0}{\pi} iacent; per symmetriam angvlorvm complementarivm sinvs angvli est cosinvs complementi eivs, et cvm cosinvs in [0,π]\icc{0}{\pi} decrescat, sinvs in [−π2,π2]\icc{-\frac{\pi}{2}}{\frac{\pi}{2}} crescit.

  4. tan⁡α=sin⁡αcos⁡α<sin⁡βcos⁡α<sin⁡βcos⁡β=tan⁡β\tan \alpha = \frac{\sin \alpha}{\cos \alpha} < \frac{\resaltar{\sin \beta}}{\cos \alpha} < \frac{\sin \beta}{\resaltar{\cos \beta}} = \dato{t}{\tan \beta}

    Pro tangente sint 0≤α<β<π20 \leq \alpha < \beta < \frac{\pi}{2}. Per gradvs svperiores, 0≤sin⁡α<sin⁡β0 \leq \sin \alpha < \sin \beta et cos⁡α>cos⁡β>cos⁡π2=0\cos \alpha > \cos \beta > \cos \frac{\pi}{2} = 0. Si primvm nvmerator avgetvr et deinde denominator minvitvr, ambo positivi, qvotiens avgetvr.

  5. tan⁡α=−tan⁡(−α)<−tan⁡(−β)=tan⁡β\tan \alpha = \resaltar{-\tan(-\alpha)} < \resaltar{-\tan(-\beta)} = \tan \beta

    Tangens impar est, qvia est qvotiens sinvs, imparis, per cosinvm, parem. Si −π2<α<β≤0-\frac{\pi}{2} < \alpha < \beta \leq 0, opposita 0≤−β<−α0 \leq -\beta < -\alpha satisfacivnt, et signo mvtato inaeqvalitas gradvs svperioris invertitvr; si α<0<β\alpha < 0 < \beta, tangens in α\alpha negativa est et in β\beta positiva. Ergo tangens in toto intervallo (−π2,π2)\ioo{-\frac{\pi}{2}}{\frac{\pi}{2}} crescit.

Restrictiones biiectivae

Omnis fvnctio stricte monotona iniectiva est, secvndvm theorema fvnctionvm monotonarvm ex lectione de operationibvs fvnctionvm. Restat vt qvaeqve restrictio omnes valores copiae imaginvm svae capiat.

TheoremaRestrictiones biiectivae

Sinvs, ad [−π2,π2]\icc{-\frac{\pi}{2}}{\frac{\pi}{2}} restrictvs, est biiectio in [−1,1]\icc{-1}{1}; cosinvs, ad [0,π]\icc{0}{\pi} restrictvs, est biiectio in [−1,1]\icc{-1}{1}, et tangens, ad (−π2,π2)\ioo{-\frac{\pi}{2}}{\frac{\pi}{2}} restricta, est biiectio in R\mathbb{R}.

Iniectivitas ex theoremate svperiore seqvitvr. Qvod qvaeqve restrictio omnes svos valores capiat, seqvitvr ex proprietate operimenti axiomatis percvrsvs, secvndvm qvam omne pvnctvm circvmferentiae vnitariae est extremvm arcvs qvi a (1,0)(1, 0) incipit, et ex lemmate secvndvm qvod vna conversio pvncta non iterat, ambobvs ex lectione de angvlis eorvmqve mensvra.

Demonstratio

  1. 0≤α<β≤π→cos⁡α≠cos⁡β0 \leq \alpha < \beta \leq \pi \rightarrow \resaltar{\cos \alpha \neq \cos \beta}

    Per theorema svperivs, tres restrictiones stricte monotonae svnt, et omnis fvnctio stricte monotona iniectiva est: dvo angvli distincti semicircvli svperioris abscissas distinctas habent.

  2. Q=(c,1−c2),c2+(1−c2)2=1→Q=P(θ)Q = \left( c, \sqrt{1 - c^2} \right), \quad \resaltar{c^2 + \left( \sqrt{1 - c^2} \right)^2 = 1} \rightarrow \dato{q}{Q = P(\theta)}

    Sit cc cvm −1≤c≤1-1 \leq c \leq 1. Pvnctvm QQ qvod infra scribitvr in circvmferentia vnitaria iacet; per operimentvm et qvia vna conversio pvncta non iterat, est P(θ)P(\theta) pro vnico θ\theta in [0,2π)\ico{0}{2\pi}.

  3. sin⁡θ=1−c2≥0→θ∈[0,π],cos⁡θ=c\sin \theta = \sqrt{1 - c^2} \geq 0 \rightarrow \theta \in \icc{0}{\pi}, \quad \dato{c}{\cos \theta = c}

    Ordinata pvncti QQ negativa non est, et sinvs in (π,2π)\ioo{\pi}{2\pi} negativvs est, secvndvm signa per qvadrantes ex lectione de rationibvs trigonometricis: illvd θ\theta in [0,π]\icc{0}{\pi} iacet, et cosinvs eivs est cc. Cosinvs restrictvs omnes valores intervalli [−1,1]\icc{-1}{1} capit.

  4. sin⁡(π2−θ)=cos⁡θ=c,π2−θ∈[−π2,π2]\sin\left( \frac{\pi}{2} - \theta \right) = \resaltar{\cos \theta} = \resaltar{c}, \quad \frac{\pi}{2} - \theta \in \icc{-\frac{\pi}{2}}{\frac{\pi}{2}}

    Per symmetriam angvlorvm complementarivm sinvs angvli π2−θ\frac{\pi}{2} - \theta est cosinvs angvli θ\theta, et dvm θ\theta intervallvm [0,π]\icc{0}{\pi} percvrrit, complementvm eivs [−π2,π2]\icc{-\frac{\pi}{2}}{\frac{\pi}{2}} percvrrit: sinvs qvoqve restrictvs omnes valores intervalli [−1,1]\icc{-1}{1} capit.

  5. tan⁡x=m,x−kπ∈(−π2,π2),tan⁡(x−kπ)=m\tan x = m, \quad x - k\pi \in \ioo{-\frac{\pi}{2}}{\frac{\pi}{2}}, \quad \tan(x - k\pi) = m

    Pro tangente theorema proprietatvm eivs, ex lectione de fvnctionibvs trigonometricis earvmqve delineationibvs, dicit copiam imaginvm eivs esse R\mathbb{R}: dato mm, recta declivitatis mm per originem dvcta semicircvlvm dextrvm secat in pvncto P(x)P(x) cvm xx inter −π2-\frac{\pi}{2} et π2\frac{\pi}{2}. Etiam per periodicitatem videtvr: si tan⁡x=m\tan x = m, svbtrahendo ab xx mvltiplvm integrvm ipsivs π\pi, vt in redvctione ad primam conversionem, obtinetvr nvmervs eivs intervalli eandem tangentem habens.