Formule trigonometrice a b a b c b a c - Math

[Pages:5]Formule trigonometrice

a

b

a

b

1. sin = ; cos = ; tg = ; ctg = ;

c

c

b

a

(a, b - catetele, c - ipotenuza triunghiului dreptunghic, - unghiul, opus catetei a).

sin

cos

2. tg = ; ctg = .

cos

sin

3. tg ctg = 1.

4. sin ? = cos ; sin( ? ) = sin .

2

5. cos ? = sin ; cos( ? ) = - cos .

2

6. tg 2 ? = ctg ; ctg 2 ? = tg .

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 . 2 tg

17. tg 2 = 1 - tg2 .

ctg2 - 1

18. ctg 2 =

.

2 ctg

19. sin 3 = 3 sin - 4 sin3 .

20. cos 3 = 4 cos3 - 3 cos .

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1 - cos

21. sin =

.

2

2

1 + cos

22. cos =

.

2

2

1 - cos

23. tg =

.

2

1 + cos

sin 1 - cos

24. tg =

=

.

2 1 + cos sin

1 + cos

25. ctg =

.

2

1 - cos

sin 1 + cos

26. ctg =

=

.

2 1 - cos sin

27.

1

+

cos

=

2

cos2

.

2

28.

1

-

cos

=

2

sin2

.

2

29. sin ? sin = 2 sin ? cos .

2

2

+ -

30. cos + cos = 2 cos

cos

.

2

2

31.

cos - cos

+ = -2 sin

sin - .

2

2

sin( ? )

32.

tg ? tg

=

. cos cos

33. ctg ? ctg = sin( ? ). sin sin

1 34. sin sin = 2[cos( - ) - cos( + )].

1 35. sin cos = [sin( + ) + sin( - )].

2

1 36. cos cos = 2[cos( + ) + cos( - )].

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37. Ecuatii trigonometrice elementare:

sin x = a, cos x = a,

|a| 1; |a| 1;

x = (-1)n arcsin a + n; x = ? arccos a + 2n;

tg x = a, x = arctg a + n; ctg x = a, x = arcctg a + n

n Z.

38. arcsin x + arccos x = , 2

|x| 1.

39. arctg x + arcctg x = .

2

40. arcsec x + arccosec x = , |x| 1.

2

41. sin(arcsin x) = x, x [-1; +1].

42. arcsin(sin x) = x, x - ; .

22

43. cos(arccos x) = x, x [-1; +1].

44. arccos(cos x) = x, x [0; ].

45. tg(arctg x) = x, x R.

46. arctg(tg x) = x, x - 2 ; 2 . 47. ctg(arcctg x) = x, x R.

48. arcctg(ctg x) = x, x (0; ).

x

49. arcsin x = arccos 1 - x2 = arctg

1 - x2

= arcctg

,

1 - x2

x

50.

arccos x

=

arcsin 1 - x2

=

arctg

1 - x2

=

arcctg

x

,

x

1 - x2

0 < x < 1. 0 < x < 1.

51. arctg x = arcsin x

= arccos 1

1 = arcctg , 0 < x < +.

1 + x2

1 + x2

x

52. arcctg x = arcsin 1

= arccos x

1 = arctg , 0 < x < +.

1 + x2

1 + x2

x

arcsin(x 1 - y2 + y 1 - x2),

53.

arcsin x+arcsin y =

- arcsin(x 1 - y2 + y 1 - x2),

- - arcsin(x 1 - y2 + y 1 - x2),

daca xy 0 sau x2 + y2 1; daca x > 0, y > 0 si x2 + y2 > 1; daca x < 0, y < 0 si x2 + y2 > 1.

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arcsin(x 1 - y2 - y 1 - x2),

54.

arcsin x-arcsin y

=

- arcsin(x 1 - y2 - y 1 - x2),

- - arcsin(x 1 - y2 - y 1 - x2),

daca xy 0 sau x2 + y2 1; daca x > 0, y < 0 si x2 + y2 > 1; daca x < 0, y > 0 si x2 + y2 > 1.

55. arccos x + arccos y = arccos(xy - (1 - x2)(1 - y2)),

daca x + y 0;

2 - arccos(xy - (1 - x2)(1 - y2)), daca x + y < 0.

56. arccos x - arccos y = - arccos(xy + (1 - x2)(1 - y2)), daca x y; arccos(xy + (1 - x2)(1 - y2)), daca x < y.

x+y

57.

arctg x + arctg y =

arctg

,

1 - xy

x+y

+ arctg

,

1 - xy

x+y

- + arctg

,

1 - xy

daca xy < 1; daca x > 0 si xy > 1; daca x < 0 si xy > 1.

x-y

58.

arctg x - arctg y

=

arctg

,

1 + xy

+ arctg x - y , 1 + xy

x-y - + arctg

,

1 + xy

daca xy > -1; daca x > 0 si xy < -1; daca x < 0 si xy < -1.

59.

2 arcsin x =

arcsin(2x 1 - x2),

- arcsin(2x 1 - x2),

- - arcsin(2x 1 - x2),

2

daca |x| ;

2

2

daca < x 1;

2

2

daca - 1 x < - 2 .

arccos(2x2 - 1)

cand 0 x 1;

60. 2 arccos x =

2 - arccos(2x2 - 1) cand - 1 x < 0.

2x

61.

2 arctg x =

arctg 1 - x2 , 2x

+ arctg 1 - x2 , 2x

- + arctg 1 - x2 ,

daca |x| < 1; daca x > 1; daca x < -1.

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62.

1 2

arcsin

x

=

arcsin

1 - 1 - x2 ,

2 1 - 1 - x2

- arcsin

,

2

daca 0 x 1; daca - 1 x < 0.

1

1+x

63. arccos x = arccos

, daca - 1 x 1.

2

2

1 + x2 - 1

64.

1 2

arctg

x

=

arctg

x

, daca x = 0;

0, daca x = 0.

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