FORMULE

1. sin²t+cos²t=1 ;t=2k¶+t* ;t*€(0,2¶ )2. sin t=sin(2k¶+t)3. cos t=cos(2k¶+t)4. sin(a±b)=sin a cos b± sin b cos a5. cos(a+b)=cos a cos b-sin a sin b6. cos(a-b)=cos a cos b+sin a sin b7. cos(¶/2-t)=sin t8. sin(¶/2-t)=cos t9. cos (-t)=cos t10. sin(-t)= -sin t11. cos (¶-x)= -cos x12. sin(¶-x)=sin x13. sin(¶+x)= -sin x14. sin 2x=2sin x cos x15. cos 2x=cos² x-sin² x =2cos² x-1=1-2sin² x16. sin 3x= 3sin x-4sin³ x17. cos 3x=4cos³ x-3cos x18. tg (-x)=-tg x x € R-A19. tg(¶+x)=tg x x € R-A20. ctg(-x)=-ctg x x € R-B21. ctg(¶+x)=ctg x x € R-B

Formule trigonometrice1. sin = a c; cos = b c; tg = a b; ctg = b a; (a; b - catetele, c - ipotenuza triunghiului dreptunghic, - unghiul, opus catetei a).2. tg = sin cos ; ctg = cos sin : 3. tg ctg = 1:4. sin = cos ; sin( ) = sin : 2

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5. cos = sin ; cos( ) = cos : 26. tg = ctg ; ctg = tg : 2 2 7. sec = cosec ; cosec = sec : 2 2 8. sin2 + cos2 = 1:9. 1 + tg2 = sec2 :10. 1 + ctg2 = cosec2 :11. sin( ) = sin cos sin cos :12. cos( ) = cos cos sin sin :13. tg( ) = tg tg 1 tg tg : 14. ctg( ) = ctg ctg 1 ctg ctg : 15. sin 2 = 2 sin cos :16. cos 2 = cos2 sin2 :17. tg 2 = 2 tg 1 tg2 : 18. ctg 2 = ctg2 1 2 ctg : 19. sin 3 = 3 sin 4 sin3 :20. cos 3 = 4 cos3 3 cos :0 Copyright c 1999 ONG TCV Scoala Virtuala a Tanarului Matematician http://math.ournet.md

s21. sin 2 = 1 cos 2 : s22. cos 2 = 1 + cos 2 : s23. tg 2 = 1 cos 1 + cos : 24. tg 2 = sin 1 + cos = 1 cos sin : s25. ctg 2 = 1 + cos 1 cos : 26. ctg 2 = sin 1 cos = 1 + cos sin : 27. 1 + cos = 2 cos22:28. 1 cos = 2 sin22:29. sin sin = 2 sin cos : 2 2 30. cos + cos = 2 cos + 2 cos 2 :

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31. cos cos = 2 sin + 2 sin 2 : 32. tg tg = sin( ) cos cos : 33. ctg ctg = sin( ) sin sin : 34. sin sin = 1 2[cos( ) cos( + )]:35. sin cos = 1 2[sin( + ) + sin( )]:36. cos cos = 1 2[cos( + ) + cos( )]:0 Copyright c 1999 ONG TCV Scoala Virtuala a Tanarului Matematician http://math.ournet.md

37. Ecuatii trigonometrice elementare:sin x = a; jaj 1; x = ( 1)n arcsin a + n; cos x = a; jaj 1; x = arccos a + 2 n; 9 > > >> >> =n 2 Z:tg x = a; x = arctg a + n; ctg x = a; x = arcctg a + n > > > >> > ;

38. arcsin x + arccos x = 2; jxj 1:

39. arctg x + arcctg x =2:40. arcsec x + arccosec x = 2; jxj 1: 41. sin(arcsin x) = x; x 2 [ 1; +1]:

42. arcsin(sin x) = x; x 2 : 2; 2 43. cos(arccos x) = x; x 2 [ 1; +1]: 44. arccos(cos x) = x; x 2 [0; ]: 45. tg(arctg x) = x; x 2 R:

46. arctg(tg x) = x; x 2 : 2; 2

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47. ctg(arcctg x) = x; x 2 R: 48. arcctg(ctg x) = x; x 2 (0; ): p 1 x2 = arctg x p 1 x249. arcsin x = arccos p 1 x2 = arcctg x ; 0 < x < 1: p 1 x2 = arctg p 1 x2 x 50. arccos x = arcsin x = arcctg p 1 x2 ; 0 < x < 1: x 1 51. arctg x = arcsin p 1 + x2 = arccos p 1 + x2 = arcctg1x; 0 < x < +1: 1 x 52. arcctg x = arcsin p = arccos p 1 + x2 = arctg1x; 0 < x < +1: 1 + x22 arcsin(x p 1 y2 + y p 1 x2); daca xy 0 sau x2 + y2 1; 6 p 1 y2 + y p 1 x2);653. arcsin x+arcsin y = 6 arcsin(x daca x > 0; y > 0 si x2 + y2 > 1;4arcsin(x p 1 y2 + y p 1 x2); daca x < 0; y < 0 si x2 + y2 > 1:0 Copyright c 1999 ONG TCV Scoala Virtuala a Tanarului Matematician http://math.ournet.md

2 arcsin(x p 1 y2 y p 1 x2); daca xy 0 sau x2 + y2 1;

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6 p p 1 x2);654. arcsin x arcsin y = 6 arcsin(x 1 y2 y daca x > 0; y < 0 si x2 + y2 > 1;4arcsin(x p 1 y2 y p 1 x2); daca x < 0; y > 0 si x2 + y2 > 1:2 q arccos(xy (1 x2)(1 y2)); daca x + y 0; 655. arccos x + arccos y = 4 q 2 arccos(xy (1 x2)(1 y2)); daca x + y < 0: 2 q arccos(xy + (1 x2)(1 y2)); daca x y; 656. arccos x arccos y = 4 q arccos(xy + (1 x2)(1 y2)); daca x < y: 266 arctg x + y 1 xy; daca xy < 1; 657. arctg x + arctg y = 6 6 6 6 + arctg x + y 1 xy; daca x > 0 si xy > 1; 64 + arctg x + y 1 xy ; daca x < 0 si xy > 1: 266 arctg x y 1 + xy; daca xy > 1; 658. arctg x arctg y = 6 6 6 6 + arctg x y 1 + xy; daca x > 0 si xy < 1; 64 + arctg x y 1 + xy ; daca x < 0 si xy < 1: 2 p p 2 6 arcsin(2x 1 x2); daca jxj ; 6 2 6 p 2 6 p

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659. 2 arcsin x = 6 arcsin(2x 1 x2); daca < x 1; 66 2 64 p 1 x2); daca 1 x < p 2 arcsin(2x : 2260. 2 arccos x = 4 arccos(2x2 1) cand 0 x 1; 2 arccos(2x2 1) cand 1 x < 0:266 arctg 2x 1 x2; daca jxj < 1; 66 2x 1 x2; 61. 2 arctg x = 6 + arctg daca x > 1; 6664 2x 1 x2 + arctg ; daca x < 1: 0 Copyright c 1999 ONG TCV Scoala Virtuala a Tanarului Matematician http://math.ournet.md

2 s p 1 x21 6 arcsin ; daca 0 x 1; 1 2arcsin x = 6 6 2 62. 6 s 6 p 4 1 1 x2 2 arcsin ; daca 1 x < 0: s63. 1 2arccos x = arccos 1 + x 2 ; daca 1 x 1: 2 p

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1 + x2 11 2arctg x = 6 arctg ; daca x 6= 0; 64. 6 x 40; daca x = 0:

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0 Copyright c 1999 ONG TCV Scoala Virtuala a Tanarului Matematician http://math.ournet.md

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