Formule trigonometrice [604474]

Formule trigonometrice
1. 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. tgctg= 1:
4. sin
2
= cos; sin() =sin:
5. cos
2
=sin; cos() =cos:
6. tg
2
=ctg; ctg
2
=tg:
7. sec
2
=cosec; cosec
2
= sec:
8. sin2+ cos2= 1:
9. 1 + tg2= sec2:
10. 1 + ctg2= cosec2:
11. sin() = sincossincos:
12. cos() = coscossinsin:
13. tg() =tgtg
1tgtg:
14. ctg() =ctgctg1
ctgctg:
15. sin 2= 2 sincos:
16. cos 2= cos2sin2:
17. tg 2=2 tg
1tg2:
18. ctg 2=ctg21
2 ctg:
19. sin 3= 3 sin4 sin3:
20. cos 3= 4 cos33 cos:
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1999 ONG TCV Scoala Virtuala a Tanarului Matematician http://math.ournet.md
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21.sin
2=s
1cos
2:
22.cos
2=s
1 + cos
2:
23.tg
2=s
1cos
1 + cos:
24. tg
2=sin
1 + cos=1cos
sin:
25.ctg
2=s
1 + cos
1cos:
26. ctg
2=sin
1cos=1 + cos
sin:
27. 1 + cos = 2 cos2
2:
28. 1cos= 2 sin2
2:
29. sinsin= 2 sin
2cos
2:
30. cos+ cos= 2 cos+
2cos
2:
31. coscos=2 sin+
2sin
2:
32. tgtg=sin()
coscos:
33. ctgctg=sin()
sinsin:
34. sinsin=1
2[cos()cos(+)]:
35. sincos=1
2[sin(+) + sin()]:
36. coscos=1
2[cos(+) + cos()]:
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1999 ONG TCV Scoala Virtuala a Tanarului Matematician http://math.ournet.md
2

37. Ecuatii trigonometrice elementare:
sinx=a;jaj1;x= (1)narcsina+n;
cosx=a;jaj1;x=arccosa+ 2n;
tgx=a; x = arctga+n;
ctgx=a; x = arcctga+n9
>>>>>>=
>>>>>>;n2Z:
38. arcsinx+ arccosx=
2;jxj1:
39. arctgx+ arcctgx=
2:
40. arcsecx+ arccosecx=
2;jxj1:
41. sin(arcsin x) =x; x2[1; +1]:
42. arcsin(sin x) =x; x2

2;
2
:
43. cos(arccos x) =x; x2[1; +1]:
44. arccos(cos x) =x; x2[0;]:
45. tg(arctg x) =x; x2R:
46. arctg(tg x) =x; x2

2;
2
:
47. ctg(arcctg x) =x; x2R:
48. arcctg(ctg x) =x; x2(0;):
49. arcsinx= arccosp
1x2= arctgxp
1x2= arcctgp
1x2
x;0<x< 1:
50. arccosx= arcsinp
1x2= arctgp
1x2
x= arcctgxp
1x2;0<x< 1:
51. arctgx= arcsinxp
1 +x2= arccos1p
1 +x2= arcctg1
x;0<x< +1:
52. arcctgx= arcsin1p
1 +x2= arccosxp
1 +x2= arctg1
x;0<x< +1:
53. arcsinx+arcsiny=2
6664arcsin(xp1y2+yp
1x2); dacaxy0 saux2+y21;
arcsin(xp1y2+yp
1x2);dacax>0; y> 0 six2+y2>1;
arcsin(xp1y2+yp
1x2);dacax<0;y< 0 six2+y2>1:
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1999 ONG TCV Scoala Virtuala a Tanarului Matematician http://math.ournet.md
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54. arcsinxarcsiny=2
6664arcsin(xp1y2yp
1x2); dacaxy0 saux2+y21;
arcsin(xp1y2yp
1x2);dacax>0;y< 0 six2+y2>1;
arcsin(xp1y2yp
1x2);dacax<0;y> 0 six2+y2>1:
55. arccosx+ arccosy=2
64arccos(xyq
(1x2)(1y2)); dacax+y0;
2arccos(xyq
(1x2)(1y2));dacax+y<0:
56. arccosxarccosy=2
64arccos(xy+q
(1x2)(1y2));dacaxy;
arccos(xy+q
(1x2)(1y2)); dacax<y:
57. arctgx+ arctgy=2
666666664arctgx+y
1xy; dacaxy< 1;
+ arctgx+y
1xy;dacax>0 sixy> 1;
+ arctgx+y
1xy;dacax<0 sixy> 1:
58. arctgxarctgy=2
666666664arctgxy
1 +xy; dacaxy>1;
+ arctgxy
1 +xy;dacax>0 sixy<1;
+ arctgxy
1 +xy;dacax<0 sixy<1:
59. 2 arcsin x=2
6666666664arcsin(2xp
1x2); dacajxjp
2
2;
arcsin(2xp
1x2);dacap
2
2<x1;
arcsin(2xp
1x2);daca1x<p
2
2:
60. 2 arccos x=2
4arccos(2×21) cand 0x1;
2arccos(2×21) cand1x<0:
61. 2 arctgx=2
666666664arctg2x
1x2; dacajxj<1;
+ arctg2x
1x2;dacax>1;
+ arctg2x
1x2;dacax<1:
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1999 ONG TCV Scoala Virtuala a Tanarului Matematician http://math.ournet.md
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62.1
2arcsinx=2
666664arcsins
1p
1x2
2; daca 0x1;
arcsins
1p
1x2
2;daca1x<0:
63.1
2arccosx= arccoss
1 +x
2;daca1x1:
64.1
2arctgx=2
664arctgp
1 +x21
x;dacax6= 0;
0;dacax= 0:
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1999 ONG TCV Scoala Virtuala a Tanarului Matematician http://math.ournet.md
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