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Aeqvationes et inaeqvationes trigonometricae

Svmmarivm

Haec lectio aeqvationes et inaeqvationes solvit in qvibvs incognita intra fvnctiones trigonometricas inest. Primvm solvtiones generales aeqvationvm sin⁡x=c\sin x = c, cos⁡x=c\cos x = c et tan⁡x=c\tan x = c in tota recta reali demonstrat, qvae solvtiones in vna conversione per fvnctiones inversas obtentas per periodicitatem extendvnt, et nvmerat qvot earvm in intervallvm datvm cadant. Deinde docet qvomodo eae aeqvationes qvae identitatibvs, factorizatione vel angvlis mvltiplicibvs solvvntvr ad has aeqvationes elementares redvcantvr, per aeqvalitatem dvorvm sinvvm vel dvorvm cosinvvm; aeqvationes lineares in sinv et cosinv per formam harmonicam solvit, cvm condicione solvbilitatis, atqve aeqvationes homogeneas, et ostendit qvomodo divisio solvtiones perdat et elevatio ad qvadratvm eas addat. Postremo inaeqvationes in vna conversione per circvmferentiam vnitariam et per tabvlas signorvm solvit, easqve periodice extendit.

Proposita discendi

  1. Solvtiones generales aeqvationvm elementarivm sinvs, cosinvs et tangentis demonstrare et adhibere, et earvm solvtiones in intervallo dato nvmerare.
  2. Aeqvationem trigonometricam ad aeqvationes elementares redvcere per identitates, factorizationem, aeqvalitatem dvorvm sinvvm vel dvorvm cosinvvm et formam harmonicam, et decernere qvando aeqvatio linearis in sinv et cosinv solvtionem habeat.
  3. Solvtiones agnoscere qvae dividendo amittvntvr et qvae ad qvadratvm elevando vel mvltiplicando addvntvr, et qvamqve solvtionem obtentam in aeqvatione originali examinare.
  4. Inaeqvationes trigonometricas in vna conversione per circvmferentiam vnitariam et per tabvlas signorvm solvere, earvmqve solvtionem generalem scribere.

Solvtiones generales aeqvationvm elementarivm

Aeqvatio trigonometrica est aeqvatio in qva incognita in argvmento fvnctionvm trigonometricarvm inest, vt 2sin⁡2x−3sin⁡x+1=02\sin^2 x - 3\sin x + 1 = 0 vel sin⁡3x=sin⁡x\sin 3x = \sin x; eam solvere est copiam omnivm nvmerorvm realivm qvi ei satisfacivnt invenire. In lectione de fvnctionibvs trigonometricis inversis demonstratvm est theorema aeqvationvm elementarivm, qvod solvtiones aeqvationvm sin⁡x=c\sin x = c et cos⁡x=c\cos x = c in [0,2π)\ico{0}{2\pi} praebet. Iam vero sinvs et cosinvs periodvm 2π2\pi habent, tangens avtem periodvm π\pi, ex lectione de fvnctionibvs trigonometricis earvmqve delineationibvs; ex qvo seqvitvr qvamqve solvtionem infinities iterari, et ad omnes describendas satis esse eas in intervallo nosse cvivs longitvdo periodvs sit. Factvm qvod ab tali intervallo ad totam rectam transire sinit varians est theorematis redvctionis ad primam conversionem, in lectione de angvlis eorvmqve mensvra demonstrati.

LemmaRedvctio ad vnam periodvm

Sint p>0p > 0 et aa nvmeri reales. Pro omni reali xx exstat vnicvs nvmervs integer kk talis vt x−kp∈[a,a+p)x - kp \in \ico{a}{a + p}.

Mvtatio scalae intervallvm [a,a+p)\ico{a}{a + p} in primam conversionem, [0,2π)\ico{0}{2\pi}, transfert, et theorema redvctionis ad primam conversionem nvmervm integrvm qvaesitvm praebet. Figvra adhibet a=1a = 1, p=52p = \frac{5}{2} et x=8x = 8.

Demonstratio

  1. θ=2πp(x−a)=θ0+2kπ,0≤θ0<2π\theta = \frac{2\pi}{p} (x - a) = \dato{t}{\theta_0 + 2k\pi, \quad 0 \leq \theta_0 < 2\pi}

    Sit θ\theta nvmervs qvi ex xx obtinetvr svbtrahendo aa et mvltiplicando per 2πp\frac{2\pi}{p}. Ex theoremate redvctionis ad primam conversionem, exstant vnicvs integer kk et vnicvm θ0\theta_0 in [0,2π)\ico{0}{2\pi} talia vt θ=θ0+2kπ\theta = \theta_0 + 2k\pi.

  2. x−kp=a+p2πθ0→a≤x−kp<a+px - kp = a + \frac{p}{2\pi} \theta_0 \rightarrow \resaltar{a \leq x - kp < a + p}

    Mvltiplicamvs per p2π\frac{p}{2\pi}, qvi positivvs est et inaeqvalitates servat, et addimvs aa: nvmervs x−kpx - kp in intervallo postvlato est. In figvra, k=2k = 2 et x−2p=3x - 2p = 3.

  3. a≤x−jp<a+p→0≤θ−2jπ<2π→j=ka \leq x - jp < a + p \rightarrow 0 \leq \resaltar{\theta - 2j\pi} < 2\pi \rightarrow \resaltar{j = k}

    Vnicitas: si integer jj qvoqve a≤x−jp<a+pa \leq x - jp < a + p implet, svbtrahendo aa et mvltiplicando per 2πp\frac{2\pi}{p} obtinetvr θ−2jπ\theta - 2j\pi in [0,2π)\ico{0}{2\pi} esse, ita vt θ=(θ−2jπ)+2jπ\theta = \left( \theta - 2j\pi \right) + 2j\pi sit alia decompositio similis ei primi gradvs; ex vnicitate qvam theorema redvctionis praebet, j=kj = k.

TheoremaSolvtiones generales

Sit cc nvmervs realis cvm ∣c∣≤1\abs{c} \leq 1. Solvtiones aeqvationis sin⁡x=c\sin x = c svnt nvmeri arcsin⁡c+2kπ\arcsin c + 2k\pi et π−arcsin⁡c+2kπ\pi - \arcsin c + 2k\pi, et aeqvationis cos⁡x=c\cos x = c nvmeri arccos⁡c+2kπ\arccos c + 2k\pi et −arccos⁡c+2kπ-\arccos c + 2k\pi, vbi kk nvmeros integros percvrrit. Pro omni reali cc, solvtiones aeqvationis tan⁡x=c\tan x = c svnt nvmeri arctan⁡c+kπ\arctan c + k\pi, cvm kk integro. Si ∣c∣>1\abs{c} > 1, aeqvationes sinvs et cosinvs solvtionem non habent.

Notatione copiarvm et connexivorvm logicorvm, theorema sic scribitvr:

sin⁡x=c↔x=arcsin⁡c+2kπ∨x=π−arcsin⁡c+2kπ,k∈Z\sin x = c \leftrightarrow x = \arcsin c + 2k\pi \vee x = \pi - \arcsin c + 2k\pi, \quad k \in \mathbb{Z}
cos⁡x=c↔x=±arccos⁡c+2kπ,tan⁡x=c↔x=arctan⁡c+kπ,k∈Z\cos x = c \leftrightarrow x = \pm \arccos c + 2k\pi, \qquad \tan x = c \leftrightarrow x = \arctan c + k\pi, \quad k \in \mathbb{Z}

Primvm in [−π2,3π2)\ico{-\frac{\pi}{2}}{\frac{3\pi}{2}} laboratvr, qvod est vna conversio, in dvo divisa: in [−π2,π2]\icc{-\frac{\pi}{2}}{\frac{\pi}{2}} sinvs stricte crescens est, ex theoremate monotoniae in intervallis principalibvs, qvod lectio de fvnctionibvs inversis ex axiomate percvrsvs per longitvdinem arcvs dedvcit, et in (π2,3π2)\ioo{\frac{\pi}{2}}{\frac{3\pi}{2}} agit symmetria angvlorvm svpplementarivm, sin⁡(π−x)=sin⁡x\sin(\pi - x) = \sin x, ex lectione de rationibvs trigonometricis. Lemma eventvm ad totam rectam extendit, et aeqvatio cosinvs ad aeqvationem sinvs redvcitvr. Figvrae adhibent c=12c = \frac{1}{2}, cvm arcsin⁡c=π6\arcsin c = \frac{\pi}{6}.

Demonstratio

  1. −π2≤x≤π2→(sin⁡x=c↔x=α)\dato{a}{-\frac{\pi}{2} \leq x \leq \frac{\pi}{2} \rightarrow \left( \sin x = c \leftrightarrow x = \alpha \right)}

    Sit α=arcsin⁡c\alpha = \arcsin c, qvod in [−π2,π2]\icc{-\frac{\pi}{2}}{\frac{\pi}{2}} est et sin⁡α=c\sin \alpha = c implet. Cvm sinvs in eo intervallo stricte crescens sit, evndem valorem ibi bis non svmit: α\alpha est vnica solvtio in primo dimidio conversionis. In circvmferentia vnitaria, P(α)P(\alpha) est pvnctvm dimidii dextri qvod in altitvdine cc iacet.

  2. sin⁡x=sin⁡(π−x)=c↔π−x=α↔x=π−α\sin x = \resaltar{\sin(\pi - x)} = c \leftrightarrow \resaltar{\pi - x = \alpha} \leftrightarrow \dato{b}{x = \pi - \alpha}

    Si xx in (π2,3π2)\ioo{\frac{\pi}{2}}{\frac{3\pi}{2}} est, angvlvs π−x\pi - x in (−π2,π2)\ioo{-\frac{\pi}{2}}{\frac{\pi}{2}} est et evndem sinvm habet ac xx. Ex gradv proxime svperiore, xx solvtio est prorsvs cvm π−x=α\pi - x = \alpha: recta horizontalis altitvdinis cc dimidivm sinistrvm secat in pvncto symmetrico P(π−α)P(\pi - \alpha).

  3. sin⁡x=sin⁡x0=c↔x=α+2kπ∨x=π−α+2kπ\sin x = \resaltar{\sin x_0} = c \leftrightarrow \dato{g}{\resaltar{x = \alpha + 2k\pi \vee x = \pi - \alpha + 2k\pi}}

    Sit nvnc xx nvmervs realis qvilibet. Ex lemmate, cvm p=2πp = 2\pi et a=−π2a = -\frac{\pi}{2}, scribitvr x=x0+2kπx = x_0 + 2k\pi cvm x0x_0 in conversione considerata, et per periodicitatem sin⁡x=sin⁡x0\sin x = \sin x_0. Itaqve xx solvtio est si et solvm si x0x_0 est α\alpha vel π−α\pi - \alpha.

  4. α=±π2→π−α=α∨π−α=α+2π\alpha = \pm \frac{\pi}{2} \rightarrow \resaltar{\pi - \alpha = \alpha} \vee \resaltar{\pi - \alpha = \alpha + 2\pi}

    Vicissim, omnis nvmervs harvm dvarvm familiarvm solvtio est, item per periodicitatem. In delineatione sinvs, recta y=cy = c cvrvam in dvobvs pvnctis in qvaqve periodo secat. Si c=1c = 1 vel c=−1c = -1, dvae familiae congrvvnt, qvia π−α\pi - \alpha ab α\alpha mvltiplo nvmeri 2π2\pi differt, et vnicvm pvnctvm intersectionis in qvaqve periodo est.

  5. π2−x=π2−arccos⁡c+2kπ∨π2−x=π2+arccos⁡c+2kπ\frac{\pi}{2} - x = \resaltar{\frac{\pi}{2} - \arccos c} + 2k\pi \vee \frac{\pi}{2} - x = \resaltar{\frac{\pi}{2} + \arccos c} + 2k\pi

    Pro cosinv adhibentvr symmetria angvlorvm complementarivm, cos⁡x=sin⁡(π2−x)\cos x = \sin\left( \frac{\pi}{2} - x \right), et theorema angvlorvm complementarivm ex lectione de fvnctionibvs inversis, arcsin⁡c=π2−arccos⁡c\arcsin c = \frac{\pi}{2} - \arccos c. Gradvm proxime svperiorem ad angvlvm π2−x\frac{\pi}{2} - x adhibendo dvae familiae obtinentvr.

  6. cos⁡x=c↔x=±arccos⁡c+2kπ\cos x = c \leftrightarrow \resaltar{x = \pm \arccos c + 2k\pi}

    Explicando xx, familiae svnt arccos⁡c−2kπ\arccos c - 2k\pi et −arccos⁡c−2kπ-\arccos c - 2k\pi; cvm −k-k nvmeros integros percvrrat dvm kk eos percvrrit, eae svnt qvae in envntiato. In circvmferentia, pvncta P(β)P(\beta) et P(−β)P(-\beta), cvm β=arccos⁡c\beta = \arccos c, svnt dvae intersectiones rectae verticalis abscissae cc.

  7. tan⁡x=tan⁡x0=c↔x=arctan⁡c+kπ\tan x = \resaltar{\tan x_0} = c \leftrightarrow \resaltar{x = \arctan c + k\pi}

    Pro tangente, periodi π\pi, lemma adhibetvr cvm p=πp = \pi et a=−π2a = -\frac{\pi}{2}: si x=x0+kπx = x_0 + k\pi cvm x0x_0 in (−π2,π2)\ioo{-\frac{\pi}{2}}{\frac{\pi}{2}}, vbi tangens est biiectio in R\mathbb{R}, vnica solvtio est x0=arctan⁡cx_0 = \arctan c; pvnctvm −π2-\frac{\pi}{2} non nvmeratvr, qvia ibi tangens definita non est. Si ∣c∣>1\abs{c} > 1, neqve sinvs neqve cosinvs valorem cc attingit, qvia eorvm valores in [−1,1]\icc{-1}{1} svnt.

Exempli gratia, cvm arcsin⁡(−12)=−π6\arcsin\left( -\frac{1}{2} \right) = -\frac{\pi}{6}, solvtiones aeqvationis sin⁡x=−12\sin x = -\frac{1}{2} svnt eae dvarvm familiarvm infra scriptarvm; cvm k=1k = 1 in prima et k=0k = 0 in altera obtinentvr dvae qvae in [0,2π)\ico{0}{2\pi} svnt, scilicet 11π6\frac{11\pi}{6} et 7π6\frac{7\pi}{6}, vt in lectione de fvnctionibvs inversis.

sin⁡x=−12↔x=−π6+2kπ∨x=7π6+2kπ,k∈Z\sin x = -\frac{1}{2} \leftrightarrow x = -\frac{\pi}{6} + 2k\pi \vee x = \frac{7\pi}{6} + 2k\pi, \quad k \in \mathbb{Z}

Solvtiones aeqvationis sin⁡x=0\sin x = 0 dvas familias efficivnt, 2kπ2k\pi et π+2kπ\pi + 2k\pi, qvae simvl svnt mvltipla integra nvmeri π\pi; eae aeqvationis cos⁡x=0\cos x = 0 svnt ±π2+2kπ\pm\frac{\pi}{2} + 2k\pi, qvae simvl svnt nvmeri π2+kπ\frac{\pi}{2} + k\pi. Hae dvae descriptiones compendiosae saepe adhibentvr:

sin⁡x=0↔x=kπ,cos⁡x=0↔x=π2+kπ,k∈Z\sin x = 0 \leftrightarrow x = k\pi, \qquad \cos x = 0 \leftrightarrow x = \frac{\pi}{2} + k\pi, \quad k \in \mathbb{Z}

Nvmervs solvtionvm in intervallo

Qvaestiones de aeqvationibvs trigonometricis plervmqve ad intervallvm restringvntvr, fere semper [0,2π)\ico{0}{2\pi}. Lemma dicit qvamqve familiam formae x1+2kπx_1 + 2k\pi vnvm prorsvs membrvm in [a,a+2π)\ico{a}{a + 2\pi} habere, qvaliscvmqve sit aa; ex qvo seqvitvr nvmervs solvtionvm aeqvationvm elementarivm in qvavis conversione.

TheoremaSolvtiones in vna conversione

Sint aa et cc nvmeri reales. Si ∣c∣<1\abs{c} < 1, vtraqve aeqvatio sin⁡x=c\sin x = c et cos⁡x=c\cos x = c dvas prorsvs solvtiones in [a,a+2π)\ico{a}{a + 2\pi} habet, et si ∣c∣=1\abs{c} = 1, vnam prorsvs. Aeqvatio tan⁡x=c\tan x = c vnam prorsvs solvtionem in [a,a+π)\ico{a}{a + \pi} habet.

Qvaeqve familia solvtionvm vnicvm membrvm conversioni affert, ex lemmate; satis est scire qvando dvae familiae sinvs, vel dvae cosinvs, vna eademqve familia sint. Figvrae adhibent a=1a = 1, cvm c=12c = \frac{1}{2} et c=1c = 1.

Demonstratio

  1. x1+2jπ∈[a,a+2π)↔j=−k\dato{u}{x_1 + 2j\pi \in \ico{a}{a + 2\pi} \leftrightarrow j = -k}

    Sit x1x_1 nvmervs realis. Ex lemmate, cvm p=2πp = 2\pi, exstat vnicvs integer kk talis vt x1−2kπx_1 - 2k\pi in [a,a+2π)\ico{a}{a + 2\pi} sit: familia nvmerorvm x1+2jπx_1 + 2j\pi, cvm jj integro, vnvm prorsvs membrvm in conversione habet.

  2. α+2kπ=π−α+2jπ↔α=π2+(j−k)π↔α=±π2↔c=±1\alpha + 2k\pi = \pi - \alpha + 2j\pi \leftrightarrow \resaltar{\alpha = \frac{\pi}{2} + (j - k)\pi} \leftrightarrow \alpha = \pm \frac{\pi}{2} \leftrightarrow \resaltar{c = \pm 1}

    Sit α=arcsin⁡c\alpha = \arcsin c, in [−π2,π2]\icc{-\frac{\pi}{2}}{\frac{\pi}{2}}. Si dvae familiae sinvs membrvm commvne habent, eadem familia svnt, idqve prorsvs accidit cvm α\alpha est π2\frac{\pi}{2} vel −π2-\frac{\pi}{2}, id est cvm c=±1c = \pm 1. Itaqve, si ∣c∣<1\abs{c} < 1 dvae familiae dvas solvtiones distinctas conversioni affervnt, et si ∣c∣=1\abs{c} = 1, vnam tantvm.

  3. β+2kπ=−β+2jπ↔β=(j−k)π↔β∈{0,π}↔c=±1\beta + 2k\pi = -\beta + 2j\pi \leftrightarrow \resaltar{\beta = (j - k)\pi} \leftrightarrow \beta \in \left\{ 0, \pi \right\} \leftrightarrow \resaltar{c = \pm 1}

    Sit β=arccos⁡c\beta = \arccos c, in [0,π]\icc{0}{\pi}. Dvae familiae cosinvs membrvm commvne prorsvs habent cvm β\beta est mvltiplvm integrvm nvmeri π\pi, id est cvm β\beta est 00 vel π\pi, qvod aeqvivalet ad c=±1c = \pm 1; nvmeratio eadem est ac pro sinv.

  4. −π2<arctan⁡c<π2→arctan⁡c+kπ≠π2+jπ-\frac{\pi}{2} < \arctan c < \frac{\pi}{2} \rightarrow \resaltar{\arctan c + k\pi \neq \frac{\pi}{2} + j\pi}

    Pro tangente vna sola familia est, cvivs membra consecvtiva intervallo π\pi distant, et lemma, cvm p=πp = \pi, ei vnvm prorsvs membrvm in [a,a+π)\ico{a}{a + \pi} dat. Id membrvm solvtio est, qvia arctan⁡c+kπ\arctan c + k\pi nvmqvam est formae π2+jπ\frac{\pi}{2} + j\pi, vbi tangens definita non est: arctan⁡c\arctan c in (−π2,π2)\ioo{-\frac{\pi}{2}}{\frac{\pi}{2}} est.