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
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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