Formule trigonometrice

1. sin α = \frac{a}{c}; cos α = \frac{b}{c}; tg α = \frac{a}{b}; ctg α = \frac{b}{a}; (a, b - catetele, c - ipotenuza triunghiului dreptunghic, α - unghiul, opus catetei a).

2. tg α = \frac{sin α}{cos α}; ctg α = \frac{cos α}{sin α}.

3. tg α ctg α = 1.

4. sin (\frac{π}{2} ± α) = cos α; sin(π ± α) = ∓ sin α.

5. cos (\frac{π}{2} ± α) = ∓ sin α; cos(π ± α) = − cos α.

6. tg (\frac{π}{2} ± α) = ∓ ctg α; ctg (\frac{π}{2} ± α) = ∓ tg α.

7. sec (\frac{π}{2} ± α) = ∓ cosec α; cosec (\frac{π}{2} ± α) = sec α.

8. sin^{2} α + cos^{2} α = 1.

9. 1 + tg^{2} α = sec^{2} α.

10. 1 + ctg^{2} α = cosec^{2} α.

11. sin(α ± β) = sin α cos β ± sin β cos α.

12. cos(α ± β) = cos α cos β ∓ sin α sin β.

13. tg(α ± β) = \frac{tg α ± tg β}{1 ∓ tg α tg β} .

14. ctg(α ± β) = \frac{ctg α ctg β ∓ 1}{ctg β ± ctg α}.

15. sin 2α = 2 sin α cos α.

16. cos 2α = cos^{2} α − sin^{2} α.

17. tg 2α = \frac{2 tg α}{1 − tg^{2} α}.

18. ctg 2α = \frac{ctg^{2} α − 1}{2 ctg α}.

19. sin 3α = 3 sin α − 4 sin^{3} α.

20. cos 3α = 4 cos^{3} α − 3 cos α.

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21. |sin \frac{α}{2}| = \sqrt{\frac{1 − cos α}{2}}.

22. |cos \frac{α}{2}| = \sqrt{\frac{1 + cos α}{2}}.

23. |tg \frac{α}{2}| = \sqrt{\frac{1 − cos α}{1 + cos α}}.

24. tg \frac{α}{2} = \frac{sin α}{1 + cos α} = \frac{1 − cos α}{sin α}.

25. |ctg \frac{α}{2}| = \sqrt{\frac{1 + cos α}{1 − cos α}}.

26. ctg \frac{α}{2} = \frac{sin α}{1 − cos α} = \frac{1 + cos α}{sin α}.

27. 1 + cos α = 2 cos^{2} \frac{α}{2}.

28. 1 − cos α = 2 sin^{2} \frac{α}{2}.

29. sin α ± sin β = 2 sin \frac{α ± β}{2} cos \frac{α ∓ β}{2} .

30. cos α + cos β = 2 cos \frac{α + β}{2} cos \frac{α − β}{2} .

31. cos α − cos β = −2 sin \frac{α + β}{2} sin \frac{α − β}{2} .

32. tg α ± tg β = \frac{sin(α ± β)}{cos α cos β}.

33. ctg α ± ctg β = \frac{sin(β ± α)}{sin α sin β}.

34. sin α sin β = \frac{1}{2}[cos(α − β) − cos(α + β)].

35. sin α cos β = \frac{1}{2}[sin(α + β) + sin(α − β)].

36. cos α cos β = \frac{1}{2}[cos(α + β) + cos(α − β)].

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37. Ecuatii trigonometrice elementare:

\begin{matrix}sin x = a, |a| ≤ 1; x = (−1)^{n} arcsin a + πn; \\ cos x = a, |a| ≤ 1; x = ± arccos a + 2πn; \\ tg x = a, x = arctg a + πn; \\ ctg x = a, x = arcctg a + πn \end{matrix} \} n ∈ Z.

38. arcsin x + arccos x = \frac{π}{2} , |x| ≤ 1.

39. arctg x + arcctg x = \frac{π}{2} .

40. arcsec x + arccosec x = \frac{π}{2} , |x| ≥ 1.

41. sin(arcsin x) = x, x ∈ [−1; +1].

42. arcsin(sin x) = x, x ∈ [−\frac{π}{2} ; \frac{π}{2} ] .

43. cos(arccos x) = x, x ∈ [−1; +1].

44. arccos(cos x) = x, x ∈ [0; π].

45. tg(arctg x) = x, x ∈ R.

46. arctg(tg x) = x, x ∈ (−\frac{π}{2} ; \frac{π}{2} ) .

47. ctg(arcctg x) = x, x ∈ R.

48. arcctg(ctg x) = x, x ∈ (0; π).

49. arcsin x = arccos \sqrt{1 − x^{2}} = arctg \frac{x}{\sqrt{1 − x^{2}}} = arcctg \frac{\sqrt{1 − x^{2}}}{x} , 0 < x < 1.

50. arccos x = arcsin \sqrt{1 − x^{2}} = arctg \frac{\sqrt{1 − x^{2}}}{x} = arcctg \frac{x}{\sqrt{1 − x^{2}}} , 0 < x < 1.

51. arctg x = arcsin \frac{x}{\sqrt{1 + x^{2}}} = arccos \frac{1}{\sqrt{1 + x^{2}}} = arcctg \frac{1}{x}, 0 < x < +∞.

52. arcctg x = arcsin \frac{1}{\sqrt{1 + x^{2}}} = arccos \frac{x}{\sqrt{1 + x^{2}}} = arctg \frac{1}{x}, 0 < x < +∞.

53. arcsin x+arcsin y = [ \begin{matrix}arcsin(x\sqrt{1 − y^{2}} + y\sqrt{1 − x^{2}}), & daca xy ≤ 0 sau x^{2} + y^{2} ≤ 1; \\ π − arcsin(x\sqrt{1 − y^{2}} + y\sqrt{1 − x^{2}}), & daca x > 0, y > 0 si x^{2} + y^{2} > 1; \\ −π − arcsin(x\sqrt{1 − y^{2}} + y\sqrt{1 − x^{2}}), & daca x < 0, y < 0 si x^{2} + y^{2} > 1. \end{matrix}

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54. arcsin x−arcsin y = [ \begin{matrix}arcsin(x\sqrt{1 − y^{2}} − y\sqrt{1 − x^{2}}), & daca xy ≥ 0 sau x^{2} + y^{2} ≤ 1; \\ π − arcsin(x\sqrt{1 − y^{2}} − y\sqrt{1 − x^{2}}), & daca x > 0, y < 0 si x^{2} + y^{2} > 1; \\ −π − arcsin(x\sqrt{1 − y^{2}} − y\sqrt{1 − x^{2}}), & daca x < 0, y > 0 si x^{2} + y^{2} > 1. \end{matrix}

55. arccos x + arccos y = [ \begin{matrix}arccos(xy − \sqrt{(1 − x^{2})(1 − y^{2})}), & daca x + y ≥ 0; \\ 2π − arccos(xy − \sqrt{(1 − x^{2})(1 − y^{2})}), & daca x + y < 0. \end{matrix}

56. arccos x − arccos y = [ \begin{matrix}− arccos(xy + \sqrt{(1 − x^{2})(1 − y^{2})}), & daca x ≥ y; \\ arccos(xy + \sqrt{(1 − x^{2})(1 − y^{2})}), & daca x < y. \end{matrix}

57. arctg x + arctg y = [ \begin{matrix}arctg \frac{x + y}{1 − xy} , & daca xy < 1; \\ π + arctg \frac{x + y}{1 − xy} , & daca x > 0 si xy > 1; \\ −π + arctg \frac{x + y}{1 − xy} , & daca x < 0 si xy > 1. \end{matrix}

58. arctg x − arctg y = [ \begin{matrix}arctg \frac{x − y}{1 + xy} , & daca xy > −1; \\ π + arctg \frac{x − y}{1 + xy} , & daca x > 0 si xy < −1; \\ −π + arctg \frac{x − y}{1 + xy} , & daca x < 0 si xy < −1. \end{matrix}

59. 2 arcsin x = [ \begin{matrix}arcsin(2x\sqrt{1 − x^{2}}), & daca |x| ≤ \frac{\sqrt{2}}{2}; \\ π − arcsin(2x\sqrt{1 − x^{2}}), & daca \frac{\sqrt{2}}{2} < x ≤ 1; \\ −π − arcsin(2x\sqrt{1 − x^{2}}), & daca − 1 ≤ x < −\frac{\sqrt{2}}{2}. \end{matrix}

60. 2 arccos x = [ \begin{matrix}arccos(2x^{2} − 1) & cand 0 ≤ x ≤ 1; \\ 2π − arccos(2x^{2} − 1) & cand − 1 ≤ x < 0. \end{matrix}

61. 2 arctg x = [ \begin{matrix}arctg \frac{2x}{1 − x^{2}} , & daca |x| < 1; \\ π + arctg \frac{2x}{1 − x^{2}} , & daca x > 1; \\ −π + arctg \frac{2x}{1 − x^{2}} , & daca x < −1. \end{matrix}

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62. \frac{1}{2} arcsin x = [ \begin{matrix}arcsin \sqrt{\frac{1 − \sqrt{1 − x^{2}}}{2}} , & daca 0 ≤ x ≤ 1; \\ − arcsin \sqrt{\frac{1 − \sqrt{1 − x^{2}}}{2}} , & daca − 1 ≤ x < 0. \end{matrix}

63. \frac{1}{2} arccos x = arccos \sqrt{\frac{1 + x}{2}}, daca − 1 ≤ x ≤ 1.

64. \frac{1}{2} arctg x = [ \begin{matrix}arctg \frac{\sqrt{1 + x^{2}} − 1}{x}, daca x ≠ 0; \\ 0, daca x = 0. \end{matrix}

0 Copyright\begin{matrix}c \\ ◯ \end{matrix}1999 ONG TCV Scoala Virtuala a Tanarului Matematician http://math.ournet.md