ࡱ>   5@ bjbj22 ",XX~tz<~Np4݀߀߀߀߀߀߀$̇RoooІ///o ݀/o݀///ug i 0+7!)i: k0i:+ZtiijLPL6/,qPPP~~{B{y/ ~~BEcuatii si inecuatii trigonometrice I. Ecuatii trigonometrice Ecuatiile ce contin necunoscute sub semnul functiilor trigonometrice se numesc ecuatii trigonometrice. Cele mai simple ecuatii trigonometrice sunt ecuatiile de tipul sinx = a, cosx = a, tgx = a, ctgx = a, a R. (1)Cum rezolvarea ecuatiilor trigonometrice se reduce la rezolvarea ecuatiilor de tipul ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec1" 1) (utilizand diferite transformari), vom aminti afirmatiile de baza referitor solutiile ecuatiilor ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec1" 1). Afirmatia 1. Ecuatia sinx = a, a R, (2)pentru |a| > 1 solutii nu are, iar pentru |a| 1 multimea solutiilor ei se contine in formula x = (-1)narcsina + pn, n Z, (3)unde arcsina [-[(p)/ 2];[(p)/ 2]] este unghiul, sinusul caruia este egal cu a, iar Z desemneaza multimea numerelor intregi, sau, echivalent (tinand seama de paritatea lui n), in totaliatea  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t1x.gif" \* MERGEFORMATINET  x = arcsina + 2pk, k Z. x = p - arcsina + 2pk, (4)Nota 1. Daca in ecuatia ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec2" 2) a {0;-1;1} solutiile ei ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec3" 3) se scriu mai simplu, si anume sinx = 0 x = pn, n Z, sinx = 1 x = p/2 + 2pn, n Z, sinx = -1 x = -p/2 + 2pn, n Z. Exemplul 1. Sa se rezolve ecuatiile  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige0x.gif" \* MERGEFORMATINET  Rezolvare. a) Cum  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige1x.gif" \* MERGEFORMATINET conform ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec3" 3) solutiile ecuatiei date sunt  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige2x.gif" \* MERGEFORMATINET  sau tinand seama ca  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige3x.gif" \* MERGEFORMATINET se obtine  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige4x.gif" \* MERGEFORMATINET  b) Similar exemplului a) se obtine  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige5x.gif" \* MERGEFORMATINET sau, tinand seama arcsinus ca functia este o functie impara,  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige6x.gif" \* MERGEFORMATINET  c) Cum  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige7x.gif" \* MERGEFORMATINET rezulta ca ecuatia data nu are solutii. Afirmatia 2. Ecuatia cosx = a (5)pentru |a| > 1 nu are solutii, iar pentru |a| 1 multimea solutiilor ei se contine in formula x = arccosa + 2pn, n Z, (6)unde arccosa [0;p] este unghiul, cosinusul caruia este egal cu a. Nota 2. Daca in ecuatia ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec5" 5) a {0;1;-1} solutiile ei ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec6" 6) se scriu mai simplu, si anume cosx = 0 x = p/2 + pn, n Z, cosx = 1 x = 2pn, n Z, cosx = -1 x = p + 2pn, n Z. Exemplul 2. Sa se rezolve ecuatiile: a) cosx = -1/2; b) cosx = 2/3; c)  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige8x.gif" \* MERGEFORMATINET  Rezolvare. a) Cum  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige9x.gif" \* MERGEFORMATINET conform ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec6" 6) solutiile ecuatiei date sunt  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige10x.gif" \* MERGEFORMATINET sau tinand seama ca  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige11x.gif" \* MERGEFORMATINET se obtine  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige12x.gif" \* MERGEFORMATINET  b) Similar exemplului a) se obtine  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige13x.gif" \* MERGEFORMATINET  c) Cum  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige14x.gif" \* MERGEFORMATINET ecuatia data nu are solutii. Afirmatia 3. Ecuatia tgx = a, a R (7)are solutiile x = arctga + pn, n Z, (8)unde arctga (-p/2;p/2) este unghiul, tangenta caruia este egala cu a. Afirmatia 4. Ecuatia ctgx = a, a R (9)are solutiile x = arcctga + pn, n Z, (10)unde arcctga (0;p) este unghiul, cotangenta caruia este egala cu a. Exemplul 3. Sa se rezolve ecuatiile a) tgx = 1; b) tgx = -2; c) ctgx = -1; d) ctgx = 3. Rezolvare. a) Conform ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec8" 8) solutiile ecuatiei date sunt x = arctg1 + pn, n Z, sau tinand seama ca  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige15x.gif" \* MERGEFORMATINET se obtine  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige16x.gif" \* MERGEFORMATINET  b) Similar exemplului precedent se obtine x = arctg(-2) + pn, n Z, sau tinand seama ca arctangenta este o functie impara, x = -arctg2 + pn, n Z. c) Se tine seama de ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec10" 10) si se obtine x = arcctg (-1) + pn, n Z, sau, cum  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige17x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige18x.gif" \* MERGEFORMATINET  d) Similar exemplului c) se obtine x = arcctg3 + pn, n Z. Observatie. Ecuatiile sin f(x) = a, cos f(x) = a, tg f(x) = a, ctg f(x) = a (11)prin intermediul substitutiei f(x) = t se reduc la rezolvarea ecuatiilor ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec1" 1). Exemplul 4. Sa se rezolve ecuatiile a) sin(2x - 1) = 1; b) cos(x2 + 4) = -1; c)  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige19x.gif" \* MERGEFORMATINET  d) ctgx3 = -2. Rezolvare. a) sin(2x - 1) = 1  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t0x.gif" \* MERGEFORMATINET  sint = 1, t = 2x - 1, 2x - 1 = p/2 + 2pn, n Z  2x = p/2 + 2pn + 1, n Z x = p/4 + pn + 1/2, n Z. b) cos(x2 + 4) = -1  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t0x.gif" \* MERGEFORMATINET  cost = -1, t = x2 + 4,  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t0x.gif" \* MERGEFORMATINET  x2 + 4 = p + 2pn, n Z, p + 2pn 4,  x2 = p + 2pn - 4, n = 1,2,3,...  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige20x.gif" \* MERGEFORMATINET  n = 1,2,3,... (se tine seama ca radicalul de ordin par exista doar din valori nenegative). c)  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige21x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige22x.gif" \* MERGEFORMATINET  2x = p/3 + pn, n Z  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige230x.gif" \* MERGEFORMATINET  d) ctgx3 = -2 x3 = arcctg(-2) + pn, n Z  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige23x.gif" \* MERGEFORMATINET  Ecuatii trigonometrice reductibile la ecuatii de gradul al doilea Ecuatia asin2x + bsinx + c = 0, a, b, c R, a 0 (12)prin intermediul substitutiei t = sinx, (|t| 1) se reduce la ecuatia patrata at2 + bt + c = 0. Exemplul 5. Sa se rezolve ecuatiile a) 2sin2x - 5sinx + 2 = 0; b) sin22x - sin2x = 0; c) sin2x - sinx + 6 = 0. Rezolvare. a) Se noteaza sinx = t si ecuatia devine 2t2 - 5t + 2 = 0, de unde t1 = 1/2 si t2 = 2. Cum |t| 1, ramane t = 1/2 si prin urmare ecuatia initiala este echivalenta cu ecuatia sinx = 1/2, solutiile careia sunt (a se vedea ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec3" 3))  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige231x.gif" \* MERGEFORMATINET  b) Se noteaza sinx = t si se obtine ecuatia patrata t2 - t = 0 cu solutiile t1 = 0 si t2 = 1. Astfel ecuatia initiala este echivalenta cu totalitatea de ecuatii  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t1x.gif" \* MERGEFORMATINET sin2x = 0,sin2x = 1,de unde  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige24x.gif" \* MERGEFORMATINET  c) Similar exemplelor precedente se obtine ecuatia patrata t2 - t + 6 = 0, care nu are solutii. Rezulta ca si ecuatia trigonometrica nu are solutii. Ecuatiile acos2x + bcosx + c = 0, (13) atg2x + btgx + c = 0, (14) actg2x + bctgx + c = 0, (15)unde a, b, c R, a 0 se rezolva similar ecuatiei ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec12" 12). In cazul ecuatiei ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec13" 13) se tine seama ca t = cosx in modul urmeaza sa nu intreaca unu, iar pentru t = tgx (t = ctgx) in ecuatia ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec14" 14) (respectiv ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec15" 15)) restrictii nu sunt. Exemplul 6. Sa se rezolve ecuatiile a) 6cos2x - 5cosx + 1 = 0; b) tg22x - 4tg2x + 3 = 0; c)  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige25x.gif" \* MERGEFORMATINET  Rezolvare. a) Se noteaza cosx = t si se obtine ecuatia patrata 6t2 - 5t + 1 = 0 cu solutiile t = 1/3 si t2 = 1/2. Cum ambele solutii verifica conditia |t| 1 se obtine totalitatea  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t1x.gif" \* MERGEFORMATINET cosx = 1/3,cosx = 1/2,de unde  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige26x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige27x.gif" \* MERGEFORMATINET  b) Se noteaza tg2x = t si se obtine ecuatia patrata t2 - 4t + 3 = 0 cu solutiile t1 = 1 si t2 = 3. Prin urmare  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t1x.gif" \* MERGEFORMATINET tg2x = 1,  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t1x.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige28x.gif" \* MERGEFORMATINET tg2x = 3,2x = arctg3 + pk, k Z,de unde  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige29x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige30x.gif" \* MERGEFORMATINET  c) Se rezolva similar exemplului precedent si se obtine  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige31x.gif" \* MERGEFORMATINET  x = 2arcctg2 + 2pk, n, k Z. Ecuatia acos2x + bsinx + c = 0, (16)utilizand identitatea trigonometrica de baza sin2x + cos2x = 1, se reduce la rezolvarea unei ecuatii de tipul ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec12" 12): a(1 - sin2x) + bsinx + c = 0. Similar, ecuatia asin2x + bcosx + c = 0 (17)se reduce la rezolvarea unei ecuatii de tipul ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec13" 13): a(1 - cos2x) + bcosx + c = 0. Utilizand formulele cos2x = 1 - 2sin2x, cos2x = 2cos2x - 1 ecuatiile acos2x + bsinx + c = 0, (18) acos2x + bcosx + c = 0, (19)se reduc la rezolvarea ecuatiilor de tipul ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec12" 12) si respectiv ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec13" 13). Exemplul 7. Sa se rezolve ecuatiile: a) 2sin2x + 5cosx - 5 = 0; b)  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige32x.gif" \* MERGEFORMATINET  Rezolvare. a) Cum sin2x = 1 - cos2x, ecuatia devine 2(1 - cos2x) + 5cosx - 5 = 0 sau 2cos2x - 5cosx + 3 = 0, de unde cosx = 3/2 (aceasta ecuatie nu are solutii) sau cosx = 1, cu solutiile x = 2pk, k Z. b) Cum cos4x = 1 - 2sin22x, ecuatia devine  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige33x.gif" \* MERGEFORMATINET  sau  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige34x.gif" \* MERGEFORMATINET  de unde  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t1x.gif" \* MERGEFORMATINET sin2x = 0, INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige35x.gif" \* MERGEFORMATINET si  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige36x.gif" \* MERGEFORMATINET  Ecuatia atgx + bctgx + c = 0 (20)tinand seama ca tgxctgx = 1 ( INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige37x.gif" \* MERGEFORMATINET ) prin intermediul substitutiei t = tgx (atunci ctgx = 1/t) se reduce la o ecuatie trigonometrica de tipul ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec14" 14). Exemplul 8. Sa se rezolve ecuatia:  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige38x.gif" \* MERGEFORMATINET  Rezolvare. Cum  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige39x.gif" \* MERGEFORMATINET si  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige40x.gif" \* MERGEFORMATINET ecuatia devine tgx + 5ctgx - 6 = 0. Se noteaza tgx = t, atunci  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige41x.gif" \* MERGEFORMATINET si se obtine ecuatia patrata t2 - 6t + 5 = 0 cu solutiile t1 = 1 si t2 = 5. Asadar  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t1x.gif" \* MERGEFORMATINET tgx = 1,  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t1x.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige42x.gif" \* MERGEFORMATINET tgx = 5,x = arctg5 + pn, n Z.Ecuatii omogene Ecuatia a0sinnx + a1sinn-1xcosx + ... + ak-1sinxcosn-1x + ancosnx = 0, (21)unde a0an 0, se numeste ecuatie omogena de gradul n in raport cu sinx si cosx. Cum  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige43x.gif" \* MERGEFORMATINET nu verifica ecuatia ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec21" 21) (toti termenii, incepand cu al doilea sunt nuli, iar primul este diferit de zero) multiplicand ecuatia cu  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige44x.gif" \* MERGEFORMATINET se obtine ecuatia echivalenta a0tgnx + a1tgn-1x + ... + an-1tgx + an = 0 care prin substitutia tgx = t, se reduce la rezolvarea unei ecuatii algebrice de gradul n. Exemplul 9. Sa se rezolve ecuatiile a) sin2x - cos2x = 0;c) 5sin2x + 5sinxcosx = 3; b) sin2x + sin2x - 3cos2x = 0; d)  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige45x.gif" \* MERGEFORMATINET Rezolvare. a) Ecuatia a) reprezinta o ecuatie trigonometrica omogena de gradul intai. Se multiplica cu  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige46x.gif" \* MERGEFORMATINET si se obtine ecuatia liniara in raport cu tg2x tg2x - 1 = 0 de unde tg2x = 1 si  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige47x.gif" \* MERGEFORMATINET  b) Cum sin2x = 2sinxcosx ecuatia b) se scrie sin2x + 2sinxcosx - 3cos2x = 0 si reprezinta o ecuatie trigonometrica omogena de gradul al doilea. Se multiplica cu  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige48x.gif" \* MERGEFORMATINET si se obtine ecuatia patrata tg2x + 2tgx - 3 = 0 cu solutiile tgx = -3 si tgx = 1. Prin urmare  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t1x.gif" \* MERGEFORMATINET x = -arctg3 + pn, n Z, INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige49x.gif" \* MERGEFORMATINET c) Se scrie 3 = 31 = 3(sin2x + cos2x) si ecuatia devine 5sin2x + 5sinxcosx = 3sin2x + 3cos2x sau 2sin2x + 5sinxcosx - 3cos2x = 0 adica o ecuatie trigonometrica omogena de gradul al doilea. Se rezolva similar exemplelor precedente si se obtin solutiile x = -arctg3 + pk, k Z si  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige50x.gif" \* MERGEFORMATINET  d) Cum cos2x = cos2x - sin2x, sin2x = 2sinxcosx,  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige51x.gif" \* MERGEFORMATINET ecuatia devine  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige52x.gif" \* MERGEFORMATINET  sau  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige53x.gif" \* MERGEFORMATINET  adica este o ecuatie trigonometrica omogena de gradul al doilea. Se multiplica cu  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige48x.gif" \* MERGEFORMATINET si se obtine ecuatia patrata  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige55x.gif" \* MERGEFORMATINET  cu solutia  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige56x.gif" \* MERGEFORMATINET sau, rationalizand numitorul,  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige57x.gif" \* MERGEFORMATINET  Asadar,  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige58x.gif" \* MERGEFORMATINET  Metoda transformarii sumei functiilor trigonometrice in produs. Ecuatiile de forma sina(x) sinb(x) = 0 (22) cosa(x) cosb(x) = 0 (23)cu ajutorul formulelor transformarii sumei in produs  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige59x.gif" \* MERGEFORMATINET (24)  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige60x.gif" \* MERGEFORMATINET (25)  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige61x.gif" \* MERGEFORMATINET (26)se reduc la ecuatii trigonometrice simple. Exemplul 10. Sa se rezolve ecuatiile a) sin3x + sinx = 0; c) cos5x = sin3x;b) cosx + cos3x = 0;d) sinx + cos2x + sin3x + cos4x = 0.Rezolvare. a) sin3x + sinx = 0  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige62x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t1x.gif" \* MERGEFORMATINET  sin2x = 0, cosx = 0,  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige64x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige67x.gif" \* MERGEFORMATINET  (se observa ca solutiile  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige66x.gif" \* MERGEFORMATINET se contin in solutiile  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige67x.gif" \* MERGEFORMATINET - a se desena cercul trigonometric si a se depune pe el solutiile obtinute). b) cosx + cos3x = 0 2cos2xcos(-x) = 0. Cum functia cosinus este o functie para, se obtine totalitatea  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t1x.gif" \* MERGEFORMATINET cos2x = 0,cosx = 0,de unde  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige68x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige69x.gif" \* MERGEFORMATINET  c) Cum  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige70x.gif" \* MERGEFORMATINET (formulele de reducere) se obtine ecuatia  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige71x.gif" \* MERGEFORMATINET  sau  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige72x.gif" \* MERGEFORMATINET  de unde, tinand seama ca functia sinus este impara, iar functia cosinus este para, se obtine totalitatea  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige73x.gif" \* MERGEFORMATINET  sau  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige74x.gif" \* MERGEFORMATINET  d) Se grupeaza convenabil: (sinx + sin3x) + (cos2x + cos4x) = 0, se aplica formulele ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec24" 24) si ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec25" 25) si se obtine ecuatia 2sin2xcosx + 2cos3xcosx = 0 sau 2cosx(sin2x + cos3x) = 0, de unde rezulta totalitatea de ecuatii  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t1x.gif" \* MERGEFORMATINET cosx = 0,sin2x + cos3x = 0.Din prima ecuatie se obtine  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige75x.gif" \* MERGEFORMATINET Ecuatia secunda a totalitatii se rezolva similar exemplului c) si se obtine  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige76x.gif" \* MERGEFORMATINET (se contine in solutia deja obtinuta) si  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige77x.gif" \* MERGEFORMATINET Asadar solutiile ecuatiei initiale sunt  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige78x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige79x.gif" \* MERGEFORMATINET  Metoda transformarii produsului in suma (utilizarea formulelor sin(a b), cos(a b)). Exemplul 11. Sa se rezolve ecuatiile a) cosxcos2x - sinxsin2x = 1; b) cosxcos3x = cos4x. Rezolvare. a) cosxcos2x - sinxsin2x = 1 cos(x + 2x) = 1 cos3x = 1 3x = 2pk, k Z  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige80x.gif" \* MERGEFORMATINET  b) Cum  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige81x.gif" \* MERGEFORMATINET se obtine  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige82x.gif" \* MERGEFORMATINET  sau cos2x - cos4x = 0, de unde rezulta 2sin(-x)sin3x = 0. Ultima ecuatie este echivalenta cu totalitatea  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t1x.gif" \* MERGEFORMATINET sinx = 0,sin3x = 0,de unde  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige83x.gif" \* MERGEFORMATINET (solutiile primei ecuatii se contin in solutiile ecuatiei secunde). Metoda micsorarii puterii Aceasta metoda utilizeaza formulele  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige84x.gif" \* MERGEFORMATINET (27)  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige85x.gif" \* MERGEFORMATINET (28)  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige86x.gif" \* MERGEFORMATINET (29)  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige87x.gif" \* MERGEFORMATINET (30)  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige88x.gif" \* MERGEFORMATINET (31)in scopul micsorarii gradului ecuatiei ce urmeaza a fi rezolvate. Formulele ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec27" 27) si ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec28" 28) se utilizeaza si la rezolvarea ecuatiilor sin2ax + sin2bx = sin2cx + sin2dx, (32) cos2ax + cos2bx = cos2cx + cos2dx, (33)daca numerele a, b, c si d verifica una din conditiile a + b = c + d sau a - b = c - d. Exemplul 12. Sa se rezolve ecuatiile a) cos2x + cos22x + cos23x = 3/2;b) sin42x + cos42x = sin2xcos2x;c) cos6x + sin6x = cos2x.Rezolvare. a) Se utilizeaza formula ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec27" 27) si se obtine ecuatia echivalenta  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige89x.gif" \* MERGEFORMATINET  sau cos2x + cos4x + cos6x = 0. Se grupeaza convenabil si se obtine (cos2x + cos6x) + cos4x = 0 2cos4xcos2x + cos4x = 0 cos4x(2cos2x + 1) = 0  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t1x.gif" \* MERGEFORMATINET  cos4x = 0, cos2x = -1/2,   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige90x.gif" \* MERGEFORMATINET b) Cum (a se vedea ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec29" 29))  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige91x.gif" \* MERGEFORMATINET iar  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige92x.gif" \* MERGEFORMATINET ecuatia devine  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige93x.gif" \* MERGEFORMATINET  sau sin42x + sin4x - 2 = 0, de unde rezulta sin4x = 1 si  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige94x.gif" \* MERGEFORMATINET  c) Cum  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige95x.gif" \* MERGEFORMATINET ecuatia devine  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige96x.gif" \* MERGEFORMATINET  de unde rezulta totalitatea  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t1x.gif" \* MERGEFORMATINET cos2x = 1,  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t1x.gif" \* MERGEFORMATINET x = pn, n Z,cos2x = 1/3, INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige97x.gif" \* MERGEFORMATINET Ecuatii de tipul asinx + bcosx = c, abc 0. (34)Se propun urmatoarele metode de rezolvare a ecuatiilor de forma ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec34" 34): a) Reducerea la o ecuatie omogena de gradul al doilea in raport cu  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige98x.gif" \* MERGEFORMATINET si  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige99x.gif" \* MERGEFORMATINET  Se scrie  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige100x.gif" \* MERGEFORMATINET  si ecuatia ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec34" 34) devine  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige101x.gif" \* MERGEFORMATINET  - omogena de gradul 2 daca (c - b)(b + c) 0, sau, in caz contrar, se reduce la rezolvarea unei ecuatii omogene de gradul 1 si a unei ecuatii de tipul ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec2" 2) sau ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec5" 5). Exemplul 13. Sa se rezolve ecuatiile a) sin2x + cos2x = 1; b)  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige102x.gif" \* MERGEFORMATINET  Rezolvare. a) sin2x + cos2x = 1 2sinxcosx + cos2x - sin2x = sin2x + cos2x  2sinxcosx - 2sin2x = 0 2sinx(cosx - sinx) = 0   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t1x.gif" \* MERGEFORMATINET  sinx = 0, cosx - sinx = 0,  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t1x.gif" \* MERGEFORMATINET  sinx = 0, tgx = 1,  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t1x.gif" \* MERGEFORMATINET  x = pk, k Z,  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige116x.gif" \* MERGEFORMATINET  b)  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige103x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige104x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige105x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige104x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige104x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige106x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige104x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige104x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige107x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige108x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige104x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige104x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige109x.gif" \* MERGEFORMATINET  b) Utilizarea formulelor  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige110x.gif" \* MERGEFORMATINET (35)Cu ajutorul formulelor indicate, ecuatia ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec34" 34) se reduce al o ecuatie patrata in raport cu  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige111x.gif" \* MERGEFORMATINET Se tine seama ca aplicarea acestor formule aduce la pierderea solutiilor a = p + 2pk, k Z, din ce cauza se verifica (prin substituirea directa in ecuatia initiala), daca ele sunt sau ba solutii ale ecuatiei ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec34" 34). Exemplul 14. Sa se rezolve ecuatiile a) sin2x + cos2x = 1; b)  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige112x.gif" \* MERGEFORMATINET  Rezolvare. a) Cum  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige113x.gif" \* MERGEFORMATINET si cum  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige114x.gif" \* MERGEFORMATINET nu verifica ecuatia data, ecuatia este echivalenta cu ecuatia  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige115x.gif" \* MERGEFORMATINET  sau 1 + tg2x = 2tgx + 1 - tg2x, de unde rezulta  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t1x.gif" \* MERGEFORMATINET  tgx = 0, tgx = 1,   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t1x.gif" \* MERGEFORMATINET  x = pk, k Z,  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige116x.gif" \* MERGEFORMATINET  b) Se aplica formulele ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec35" 35 ) si se obtine  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t0x.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige117x.gif" \* MERGEFORMATINET x p + 2pk, k Z,sau  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t0x.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige118x.gif" \* MERGEFORMATINET x p + 2pk, k Z,de unde  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t0x.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige119x.gif" \* MERGEFORMATINET x p + 2pk, si  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige120x.gif" \* MERGEFORMATINET Verificarea directa arata ca si x = p + 2pk, k Z sunt solutii ale ecuatiei date. Asadar solutiile ecuatiei date sunt  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige121x.gif" \* MERGEFORMATINET  c) Metoda unghiului auxiliar. Cum abc 0 ecuatia ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec34" 34) se scrie  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige122x.gif" \* MERGEFORMATINET (36)si cum  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige123x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige124x.gif" \* MERGEFORMATINET si  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige125x.gif" \* MERGEFORMATINET rezulta ca exista un unghi a, astfel incat  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige126x.gif" \* MERGEFORMATINET  si  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige127x.gif" \* MERGEFORMATINET (37)sau un unghi b, astfel incat  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige128x.gif" \* MERGEFORMATINET  si  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige129x.gif" \* MERGEFORMATINET (38)Atunci ecuatia ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec36" 36) se scrie  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige130x.gif" \* MERGEFORMATINET  sau  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige131x.gif" \* MERGEFORMATINET  Ultimile ecuatii nu prezinta greutati in rezolvare. Nota. Se observa ca ecuatia ( HYPERLINK "http://www.math.md/school/rubrica/trigonom/trige.html" \l "ec34" 34) are solutii daca si numai daca  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige132x.gif" \* MERGEFORMATINET iar valoarea maxima a functiei f(x) = asinx + bcosx este  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige133x.gif" \* MERGEFORMATINET si valoarea minima este - INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige133x.gif" \* MERGEFORMATINET . Exemplul 15. Sa se rezolve ecuatiile a) sin2x + cos2x = 1; b) 3sinx + 4cosx = 5; c)  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige134x.gif" \* MERGEFORMATINET  Rezolvare. a) sin2x + cos2x = 1  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige135x.gif" \* MERGEFORMATINET    INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige136x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige137x.gif" \* MERGEFORMATINET    INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige138x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige139x.gif" \* MERGEFORMATINET    INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t0x.gif" \* MERGEFORMATINET  x = pn, n Z,  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige140x.gif" \* MERGEFORMATINET  b) 3sinx + 4cosx = 5  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige141x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t0x.gif" \* MERGEFORMATINET  sinxcosa + cosxsina = 1, sina = 4/5; cosa = 3/5,    INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t0x.gif" \* MERGEFORMATINET  sin(x + a) = 1, tga = 4/3,  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige142x.gif" \* MERGEFORMATINET  c) Cum valoarea maxima a membrului din stanga ecuatiei este  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige143x.gif" \* MERGEFORMATINET si  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige144x.gif" \* MERGEFORMATINET rezulta ca ecuatia nu are solutii. Ecuatii de tipul F(sinx cosx, sinxcosx) = 0. Ecuatiile de asa tip se rezolva cu ajutorul substitutiei  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige145x.gif" \* MERGEFORMATINET  Exemplul 16. 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Ckd!$$IfT634ap T$IfgdYgdY$a$gdY GHILXyz #$+, չչչչչչ՚՚{kj"hYhYB*UphhYhYB*CJaJphjshYhYB*UphhYhYB*OJQJphjIhYhYB*UphhYhY6B*]phhYhY5B*\phhYhYB*phjhYhYB*Uphj hYhYB*Uph*nprtxz|~DFHJNPVX "$&,.yyyhYhY6B*]phj6hYhYB*Uphj1hYhYB*Uphj,hYhYB*UphhYhYB*CJaJphjr'hYhYB*UphhYhYB*OJQJphhYhYB*phjhYhYB*Uph/|~"$, skQkx`kQ$IfK$Ckd|6$$IfT634ap T$IfCkdL,$$IfT634ap T  帨帊zjChYhYB*Uphj$$IfT634ap T$Ifrkd(>$IfK$L$4F: 6    34ap gQx`g QPg QZkd F$IfK$L$40u 634ap$IfK$$IfK$  (* &(.0譝jKhYhYB*UphhYhY6B*]phjmHhYhYB*UphjhYhYB*UphhYhYB*CJaJphhYhYB*OJQJphhYhYB*H*phhYhYB*phhYhYB*H*ph2 $IfK$ZkdF$IfK$L$40u 634apv&ppQ$Ifkd6G$IfK$L$\,634ap((Qx`mQ:PZkdJ$IfK$L$40{634ap$IfK$$IfK$[kdG$$IfT0 D634apT ".mQgQ $IfK$ZkdhK$IfK$L$40{634ap$IfK$^&XXv:$IfkdQ$IfK$L$r:A 634ap2%0h ! !Q!J! N Ckd\$$IfT 634ap T$If$a$gdYgdY[kdmR$$IfT0 D634apT:;%567;<  `bdfh    zxUhYhYB*CJaJphj_YhYhYB*UphhYhYB*OJQJphhYhY6B*]phhYhY5B*\phj.VhYhYB*UphjRhYhYB*UphjhYhYB*UphhYhYB*ph/) 1 - sin2x = cosx - sinx;c)  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige146x.gif" \* MERGEFORMATINET Rezolvare. a) Se noteaza t = sinx + cosx, atunci t2 = (sinx + cosx)2 = 1 + sin2x, si ecuatia devine 2t + t2 = 0, de unde t = 0 sau t = -2. Cum ecuatia sinx + cosx = -2 nu are solutii, ramane sinx + cosx = 0 - ecuatie omogena de gradul intai cu solutiile  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige147x.gif" \* MERGEFORMATINET  b) Se noteaza cosx - sinx = t, atunci sin2x = 1 - t2 si ecuatia devine t2 = t cu solutiile t = 0, t = 1. Asadar  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t1x.gif" \* MERGEFORMATINET cosx - sinx = 0,  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t1x.gif" \* MERGEFORMATINET 1 - tgx = 0, cosx - sinx = 1, INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige148x.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige149x.gif" \* MERGEFORMATINET  c) DVA al ecuatiei este  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige150x.gif" \* MERGEFORMATINET In DVA ecuatia se scrie sinx + cosx - 5sinxcosx + 1 = 0. Se noteaza t = sinx + cosx si se obtine ecuatia patrata 5t2 - 2t - 7 = 0, cu solutiile t = -1 si t = 7/5. Prin urmare sinx + cosx = -1, de unde  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige151x.gif" \* MERGEFORMATINET (nu verifica DVA al ecuatiei) sinx + cosx = 7/5, de unde  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige152x.gif" \* MERGEFORMATINET  Metoda descompunerii in factori Aceasta metoda este una din cele mai frecvente si presupune o cunoastere satisfacatoare a formulelor trigonometrice. Exemplul 17. Sa se rezolve ecuatiile a) sin3x - cos3x = cos2x; b) sin3x - sin2x + 2cosx = 2cos2x - sinx; c) 4sinx + 2cosx = 2 + 3tgx. Rezolvare. a) sin3x - cos3x = cos2x (sinx - cosx)(sin2x + sinxcosx + cos2x) = cos2x - sin2x (sinx - cosx)(1 + sinxcosx + (cosx + sinx)) = 0  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t1x.gif" \* MERGEFORMATINET  sinx - cosx = 0, 1 + sinxcosx + (cosx + sinx) = 0,   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t1x.gif" \* MERGEFORMATINET  tgx = 1,  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t0x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige153x.gif" \* MERGEFORMATINET  t = sinx + cosx,    INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t1x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige154x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t0x.gif" \* MERGEFORMATINET  t2 + 2t + 1 = 0, t = sinx + cosx,   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t1x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige155x.gif" \* MERGEFORMATINET  sinx + cosx = -1,    INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t1x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige155x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige156x.gif" \* MERGEFORMATINET  x = p + 2pm, m Z. b) Se trec toti termenii in stanga ecuatiei si se grupeaza convenabil: (sin3x + sinx) + 2cosx - (sin2x + 2cos2x) = 0. Se utilizeaza formulele sumei sinusurilor si sinusului unghiului dublu si se obtine (2sin2xcosx + 2cosx) - (2sinxcosx + 2cos2x) = 0 sau 2cosx[(sin2x + 1) - (sinx + cosx)] = 0. Se tine seama ca sin2x + 1 = 2sinxcosx + sin2x + cos2x = (sinx + cosx)2 si ecuatia devine 2cosx[(sinx + cosx)2 - (sinx + cosx)] = 0 sau 2cosx(sinx + cosx)(sinx + cosx - 1) = 0, de unde se obtine totalitatea  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t1x.gif" \* MERGEFORMATINET cosx = 0,sinx + cosx = 0,sinx + cosx - 1 = 0.Din prima ecuatie a totalitatii se obtine  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige157x.gif" \* MERGEFORMATINET Cea secunda reprezinta o ecuatie trigonometrica omogena de gradul intai cu solutiile  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige158x.gif" \* MERGEFORMATINET Ecuatia a treia se rezolva, de exemplu, prin metoda introducerii unghiului auxiliar si are solutiile x = 2pn, n Z si  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige159x.gif" \* MERGEFORMATINET Ultimul set de solutii se contine in multimea solutiilor primei ecuatii si prin urmare multimea solutiilor ecuatiei initiale este  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige160x.gif" \* MERGEFORMATINET  c) DVA al ecuatiei este  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige161x.gif" \* MERGEFORMATINET Ecuatia se scrie  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige162x.gif" \* MERGEFORMATINET  sau 4sinxcosx + 2cos2x - 2cosx - 3sinx = 0. Se grupeaza convenabil: 2cosx(2sinx - 1) + (2cos2x - 3sinx) = 0, sau, cum 2cos2x = 2(1 - sin2x) = 2 - 2sin2x, 2cosx(2sinx - 1) + (2 - 3sinx - 2sin2x) = 0. Cum 2 - 3sinx - 2sin2x = 2 - 4sinx + sinx - 2sin2x = 2(1 - 2sinx) + sinx(1 - 2sinx) = (1 - 2sinx)(2 + sinx), ecuatia devine 2cosx(2sinx - 1) + (1 - 2sinx)(2 + sinx) = 0, sau (2sinx - 1)(2cosx - sinx - 2) = 0. Cum  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige163x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige164x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige165x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige164x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige166x.gif" \* MERGEFORMATINET ecuatia se scrie  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige167x.gif" \* MERGEFORMATINET  de unde rezulta sinx = 1/2, cu solutiile  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige168x.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige169x.gif" \* MERGEFORMATINET  cu solutiile x = 2pm, m Z, INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige170x.gif" \* MERGEFORMATINET  cu solutiile  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige171x.gif" \* MERGEFORMATINET Toate solutiile obtinute verifica DVA al ecuatiei. In incheiere vom prezenta unele metode utile de rezolvare a ecuatiilor trigonometrice. Exemplul 18. Sa se rezolve ecuatiile: a) cosx + cos2x + cos3x + ... + cosnx = n, n N, n 1;b) sinx + sin2x + sin3x + ... + sinnx = n, n N, n 2;c) sin11x + cos11x = 1;d) sin10x - cos7x = 1;e)  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige172x.gif" \* MERGEFORMATINET f) 3sin2x + 4cos6xcos2x + 2sin10x = 7;g)  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige173x.gif" \* MERGEFORMATINET h) 4sin2x - 4sin23xsinx + sin23x = 0;i)  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige174x.gif" \* MERGEFORMATINET j) cosxcos2xcos4xcos8x = 1/16.Rezolvare. a) Cum pentru orice m natural |cosmx| 1, membrul din stanga ecuatiei va fi egal cu n daca si numai daca fiecare termen va fi egal cu unu. Asadar rezulta sistemul  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t0x.gif" \* MERGEFORMATINET cosx = 1,cos2x = 1,...cosnx = 1cu solutiile x = 2pk, k Z. b) Se rezolva similar exemplului a) si se obtine sistemul  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t0x.gif" \* MERGEFORMATINET sinx = 1,sin2x = 1,...sinnx = 1,care este incompatibil. Intr-adevar, solutiile primei ecuatii:  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige175x.gif" \* MERGEFORMATINET nu verifica a doua ecuatie a sistemului:  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige176x.gif" \* MERGEFORMATINET Prin urmare ecuatia nu are solutii. c) Cum sin11x sin2x, cos11x cos2x implica sin11x + cos11x sin2x + cos2x, sau sin11x + cos11x 1, iar in ultima inegalitate semnul egalitatii se atinge daca si numai daca  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t1x.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t0x.gif" \* MERGEFORMATINET sinx = 0,cosx = 1, INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t0x.gif" \* MERGEFORMATINET sinx = 1,cosx = 0.rezulta ca ecuatia are solutiile x = 2pm, m Z (din primul sistem al totalitatii) si  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige175x.gif" \* MERGEFORMATINET (din sistemul secund). d) Se utilizeaza acelasi procedeu ca si in exemplul precedent: sin10x sin2x, -cos7x cos2x, de unde sin10x - cos7x 1 si, prin urmare, semnul egalitatii se atinge cand  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t0x.gif" \* MERGEFORMATINET sin10x = sin2x,-cos7x = cos2x,adica sinx {0;-1;1}, iar cosx {0;-1}. Asadar se obtine  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige178x.gif" \* MERGEFORMATINET  e) Cum  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige179x.gif" \* MERGEFORMATINET |cos2x| 1, membrul din stanga ecuatiei va fi egal cu minus unu, daca si numai daca  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige180x.gif" \* MERGEFORMATINET  Din  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige181x.gif" \* MERGEFORMATINET rezulta x = p + 4pn si atunci cos2x = cos(2p + 8pn) = 1 -1, adica primul sistem al totalitatii este incompatibil. Din  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige182x.gif" \* MERGEFORMATINET rezulta x = -p + 4pk si atunci cos2(-p + 4pk) = cos2p = 1, deci x = -p + 4pk, k Z sunt solutiile sistemului (si ecuatiei enuntate). f) Cum 3sin2x + 4cos6xcos2x 3sin2x + 4cos2x 5 (a se vedea nota la Metoda unghiului auxiliar), 2sin10x 2 se obtine 3sin2x + 4cos6xcos2x + 2sin10x 7, si semnul egalitatii se atinge doar pentru  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t0x.gif" \* MERGEFORMATINET  |cos6x| = 1, sin10x = 1,  sau  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/t0x.gif" \* MERGEFORMATINET  sin6x = 0, sin10x = 1, de unde  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige183x.gif" \* MERGEFORMATINET  Ultimul sistem este incompatibil. In adevar  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige184x.gif" \* MERGEFORMATINET  conduce la ecuatia in numere intregi 10n = 3 + 6m sau 10n - 6m = 3 care nu are solutii: diferenta a doua numere pare nu este un numar impar. Prin urmare ecuatia enuntata nu are solutii. g) Ecuatia se scrie  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige185x.gif" \* MERGEFORMATINET  sau  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige186x.gif" \* MERGEFORMATINET  Membrul din stanga nu intrece doi (         ~                               !!!!$!%!E!F!L!M!!!!!!!!ǷשnjjmbhYhYB*UphhYhYB*H*phhYhY5B*\phju]hYhYB*UphjhYhYB*UphhYhYB*CJaJphhYhYB*phhYhY6B*]ph7    !"h###P$j$v$ e'se'n!n!K`gQK`gQgdYCkdb$$IfT 634ap T$IfCkd ]$$IfT 634ap T !!!!"""","."0"V"X"Z"`"b"~""""""`#b#d#f#n#p#|#~#####H$J$L$N$\$^$n$p$v$x$$$$$$$^%`%ɹɚhYhYB*CJaJphjhhYhYB*UphhYhYB*OJQJphjfhYhYB*UphjhYhYB*UphhYhYB*H*phhYhYB*phhYhY6B*]ph3v$x$z$$$$f%B<<<<<?$Ifkdj$$IfT4ֈiNxC634ap<T`%b%d%h%j%l%&&&&&&0&1&&&&&&&&&&&&&&&&&&&&&'' ' ''&'''0'1'4'5'6'7'H'I'O'ըըըըըըըըըըըըըՋ~ըhYhYB*H*phhYhYB*H*phj|hYhYB*UphhYhY6B*]phj`qhYhYB*UphhYhYB*CJaJphhYhYB*phjhYhYB*UphjkhYhYB*Uph1f%h%j%&&&<?4! /!4!gdY$a$gdYkdep$$IfT4ֈiNxC634ap<T$If&''](~(()5)6)a)b))!!!g !!!aCkd$$IfT634ap TCkd$$IfT634ap T$IfgdY$a$gdYgdY O'P'_'`'''''''''''''''''Y(Z([(\(](|(((() )!)")()))*)1)2)5)6)=)>)E)F)M)N)U)V)W)])^)֜hYhYB*CJaJphhYhY5B*\phjHhYhYB*UphhYhYB*H*phhYhYB*H*phj4hYhYB*UphjhYhYB*UphhYhYB*phhYhY6B*]ph2^)a)b)i)j)q)r)|)}))))))))))))***** ***,*.*:*<*B*D*P*R*T*b*d*f*r*t*v*|*~*******************+jhYhYB*UphhYhYB*OJQJphhYhYB*H*phhYhY5B*\phhYhY6B*]phhYhYB*CJaJphhYhYB*ph>))**++++,!QQK`$ O$ Zkd$IfK$L$40 634ap$IfK$$IfgdYCkd\$$IfT634ap T++++++++++++++++ , ,,,",$,*,,,,,,,,,,,,,------r.t.v.x.z.|.~..ոոթոyոjÕhYhYB*UphjShYhYB*UphjӐhYhYB*UphhYhYB*OJQJphhYhYB*CJaJphhYhY6B*]phhYhYB*phjȍhYhYB*UphjhYhYB*Uph/,,,*,,,,gQK`R=ZkdȒ$IfK$L$403 634ap$IfK$$IfZkdH$IfK$L$40 634ap,,-z.|.~..x`D  DZkd$IfK$L$40R634ap$IfK$$IfK$.................n/p/r/t/v/x/406080:0<0>0@0B000000001111 1"1׻{nhYhYB*H*phjhYhYB*UphjhYhYB*UphjhYhYB*UphjhYhYB*UphhYhY<B*phhYhYB*OJQJphhYhYB*CJaJphhYhY6B*]phhYhYB*ph+......C==Q$IfZkdš$IfK$L$403 634ap$IfK$Zkd7$IfK$L$40R634ap....v/<0\W2 TNQFK`F$IfK$$If$a$gdYkdM$$IfTr N`634ap2T<0>0@0011 1x`;Zkd0$IfK$L$40634ap$IfK$$IfK$Zkd5$IfK$L$40634ap"1.101<1>1B1D1F1H1N1P1V1X12 2 222222222222222333 3 3333333333344׋{jۮhYhYB*UphhYhY<B*phjƨhYhYB*UphjѦhYhYB*UphjhYhYB*UphhYhYB*OJQJphhYhYB*CJaJphhYhY6B*]phhYhYB*ph. 1B1D1F1H1J1V1;55gQ$IfZkdF$IfK$L$40634ap$IfK$Zkd$IfK$L$40634ap$IfK$V1222223K`CZkd$IfK$L$40634apZkd$IfK$L$40634ap$IfK$33 3 33334QV- NQFK`F$IfK$$a$gdYkd$$IfTr  b634ap2T$If4444444X5Z5\5^5`5b5d5f5l5n5v5x5z55555555555666 6(6)6162696:6;6666666666ըՙՙըՙՋըըըը~ըըըըhYhYB*H*phhYhY5B*\phhYhYB*OJQJphhYhY6B*]phjhYhYB*UphhYhYB*CJaJphhYhYB*phjhYhYB*UphjаhYhYB*Uph1444`5b5d55CQZkd$IfK$L$40 634ap$IfK$Zkd$IfK$L$40 634ap55556C6DU ?!7!$a$gdYgdY[kd$$IfT0 0 634apT$IfZkds$IfK$L$40 634apC66666S7~7777)8384858F8!!!!!!!!!K`  \kdw$$IfT40i634apT$If$a$gdYgdY6666666666666 777777$7%7&7,7-7.75767<7=7>7?7W7X7]7^7d7e7f7g7n7o7u7v7777777777777%8&8'8(8,8-838488898?8@8F8ɹhYhYB*CJaJphjhYhYB*UphjhYhYB*UphhYhYB*H*phhYhYB*phhYhY6B*]phCF8G8H8]8^8; ?:!gdY\kd$$IfT40i634apT$If\kd$$IfT40i634apTF8G8K8L8R8S8]8^8888888?9@99999 :::::":$:&:(:*:,:4:6:::::;<^<_<`<ǷǧzjjhYhYB*UphjhYhYB*UphhYhY5B*\phhYhYB*OJQJphjdhYhYB*UphjhYhYB*UphjhYhYB*UphhYhY6B*]phhYhYB*phhYhYB*CJaJph)`<a<e<h<z<{<<<<<<<M=N=O=P=Z=[=^=_=f=g=h=o=p=w=x======================= > >>>>*>+>2>3>4>?>@>F>G>N>O>P>]>^>hYhYB*H*phjhYhYB*UphjhYhYB*UphhYhY6B*]phhYhYB*phjhYhYB*UphC;b<<Q=V=====>>>>>@^AoAAAA!!!!!!!!!!!!!!XJ !7!LƷƷLCkd$$IfTy634ap T$IfgdYgdY$a$gdY^>e>f>o>p>}>~>>>>>>>>>>>>>>>>>>>V?W?X?Y?[?\???????@@ @!@#@$@@@@@@@@@@@ֶ֦֖ֆjhYhYB*UphjhYhYB*UphjhYhYB*UphjhYhYB*UphjohYhYB*UphjhYhYB*UphhYhY6B*]phhYhYB*ph4@@@ZA[A\A]ArAsAvAwAxAyAAAAAAAAAAAABBBBBBBBBBBBBB}nnhYhYB*OJQJphjhYhYB*UphhYhYB*CJaJphj3hYhYB*UphhYhYB*H*phhYhYB*H*phhYhY6B*]phjhYhYB*UphjhYhYB*UphhYhYB*ph%AABBNDODLsLe'0e'Ckdm$$IfTy634ap TCkdb$$IfTy634ap T$IfCkd$$IfTy634ap TBBBBBCCCCCCJDKDLDMDNDODqDtDDDE EEEEE%E'E*E+E1E2EFFFFFFFFFF*F,F:FL@LBLDLFLLLtMvMxMzMMMMMMMMMMMMMMM(N)NNNj,$hYhYB*UphhYhY5B*\phhYhYB*OJQJphhYhYB*CJaJphhYhY6B*]phjhYhYB*UphhYhYB*phjhYhYB*Uph6KKKKKK L?\kd#$$IfT40634apT$If\kd"$$IfT40634apT L LLLL|MMM!Q!x`5\kd&$$IfT40634apT$IfgdYgdY\kd#$$IfT40634apTMMMMMMMM?\kd'$$IfT40634apT\kd('$$IfT40634apT$IfMMM:OHQR^RhR! 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gdYskdɋ$$IfTFsN6    34apT$IfZkd>$IfK$L$40$ 634apreLCo!!!!!Q!!Q!!!hj !!L !Ʒ!6\!!k!#: !!Ϲ!! dd[$\$^gdY$dd[$\$^a$gdY & Fdd[$\$gdYgdYgdY$a$gdY(*RT\^rtz|46<>@BDFZ\^`bd  hYhYB*H*phhYhY5B*\phhYhYB*OJQJphhYhYB*phhYhY6B*]phJ $%yz{|   ɹɩəzlhYhY5B*\phjhYhYB*UphhYhYB*OJQJphjShYhYB*UphjhYhYB*Uphj\hYhYB*UphjhYhYB*UphhYhY6B*]phhYhYB*H*phhYhYB*ph*fhnprz|~$%&'+,abcduvyijhYhYB*UphjhYhYB*UphjhYhYB*Uphj hYhYB*UphjhYhYB*UphjhYhYB*UphhYhY5B*\phhYhYB*OJQJphhYhY6B*]phhYhYB*ph)tvxzDFHJ?@ABTUVWZ[op|ljhYhYB*UphhYhYB*H*phjhYhYB*UphhYhY5B*\phhYhYB*OJQJphhYhY6B*]phjhYhYB*UphjhYhYB*UphjhYhYB*UphhYhYB*ph*:C]  :dH4f!!!!!!Q!!k & Fdd[$\$gdYgdY$a$gdYgdY6789EFPQpq =>?@ xjhhYhYB*UphjhYhYB*UphjhYhYB*UphjhYhYB*UphhYhYB*OJQJphhYhY6B*]phjhYhYB*UphhYhYB*phjhYhYB*Uph/  dfhnp$&24DFNP\^np|~hYhY5B*\phhYhYB*OJQJphUhYhY6B*]phhYhYB*phIpk, k Z nu sunt solutii ale ecuatiei date (cospk = 1, cos2pk = cos4pk = cos8pk = 1) se multiplica ambii membri ai ecuatiei cu 16sinx si se utilizeaza formula sinusului unghiului dublu 16sinxcosxcos2xcos4xcos8x = sinx, 8sin2xcos2xcos4xcos8x = sinx, 4sin4xcos4xcos8x = sinx, 2sin8xcos8x = sinx, sin16x = sinx, sau sin16x - sinx = 0,  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige208x.gif" \* MERGEFORMATINET  de unde  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige209x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige210x.gif" \* MERGEFORMATINET  k 15s, s Z (deoarece x pm) si  INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige211x.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.math.md/school/rubrica/trigonom/trige212x.gif" \* MERGEFORMATINET  m Z, m 17s + 8, s Z. 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