[PDF] Formule trigonometrice a b a b c b a c - Math



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Lecture 6 : Inverse Trigonometric Functions Inverse Sine

Inverse Sine Function (arcsin x = sin 1x) The trigonometric function sinxis not one-to-one functions, hence in order to create an inverse, we must restrict its domain The restricted sine function is given by f(x) = 8



Restricted Sine Function

Inverse Sine Function (arcsin x = sin 1x) We see from the graph of the restricted sine function (or from its derivative) that the function is one-to-one and hence has an inverse, shown in red in the diagram below Hp 2,1L H-p 4,-1 2 L H1,p 2L H-1 2,-p 4 L-p 2-p 4 p 4 p 2-1 5-1 0-0 5 0 5 1 0 1 5 This inverse function, f 1(x), is denoted by f 1





Section 55 Inverse Trigonometric Functions and Their Graphs

Section 5 5 Inverse Trigonometric Functions and Their Graphs DEFINITION: The inverse sine function, denoted by sin 1 x (or arcsinx), is de ned to be the inverse of the restricted sine function



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The derivative of arcsin is given by arcsin0(x) = 1 √ 1 − x2 Proof: For x ∈ [−1,1] holds arcsin0(x) = 1 sin0 arcsin(x) = 1 cos arcsin(x) For x ∈ [−1,1] we get arcsin(x) = y ∈ hπ 2, π 2 i, and the cosine is positive in that interval, then cos(y) = + q 1 − sin2(y), hence arcsin0(x) = 1 q 1 − sin2 arcsin(x) ⇒ arcsin 0(x) = 1



Inverse trigonometric functions (Sect 76)

The derivative of arcsin is given by arcsin0(x) = 1 √ 1−x2 Proof: For x ∈ [−1,1] holds arcsin0(x) = 1 sin0 arcsin(x) = 1 cos arcsin(x) For x ∈ [−1,1] we get arcsin(x) = y ∈ hπ 2, π 2 i, and the cosine is positive in that interval, then cos(y) = + q 1−sin2(y), hence arcsin0(x) = 1 q 1−sin2 arcsin(x) ⇒ arcsin 0(x) = 1 √ 1



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2 The inverse trigonometric functions: arcsin and arccos The arcsine function is the solution to the equation: z = sinw = eiw − e−iw 2i ∗In our conventions, the real inverse tangent function, Arctan x, is a continuous single-valued function that varies smoothly from − 1 2π to +2π as x varies from −∞ to +∞ In contrast, Arccotx



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

53 arcsinx+arcsiny= 2 6 6 6 4 arcsin(x p 1 y2 + y 1 x2); daca xy 0 sau x2 + y2 1;



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Formule trigonometrice

1. sin=a

c ; cos=b c ; tg=a b ; ctg=b a (a; b- catetele,c- ipotenuza triunghiului dreptunghic,- unghiul, opus cateteia).

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. sin

2+ cos2= 1:

9. 1 + tg

2= sec2:

10. 1 + ctg

2= cosec2:

11. sin() = sincossincos:

12. cos() = coscossinsin:

13. tg() =tgtg

1tgtg:

14. ctg() =ctgctg1

ctgctg:

15. sin2= 2sincos:

16. cos2= cos2sin2:

17. tg2=2tg

1tg2:

18. ctg2=ctg21

2ctg:

19. sin3= 3sin4sin3:

20. cos3= 4cos33cos:

0

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

1 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= 2cos2

2

28. 1cos= 2sin2

2

29. sinsin= 2sin

2 cos 2

30. cos+ cos= 2cos+

2 cos 2

31. coscos=2sin+

2 sin 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()]: 0

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

2

37. Ecuatii trigonometrice elementare:

sinx=a;jaj 1;x= (1)narcsina+n; cosx=a;jaj 1;x=arccosa+ 2n; tgx=a; x= arctga+n; ctgx=a; x= arcctga+n9 >>>>>;n2Z:

38. arcsinx+ arccosx=

2 ;jxj 1:

39. arctgx+ arcctgx=

2

40. arcsecx+ arccosecx=

2 ;jxj 1:

41. sin(arcsinx) =x; x2[1;+1]:

42. arcsin(sinx) =x; x2

2 2

43. cos(arccosx) =x; x2[1;+1]:

44. arccos(cosx) =x; x2[0;]:

45. tg(arctgx) =x; x2R:

46. arctg(tgx) =x; x2

2 2

47. ctg(arcctgx) =x; x2R:

48. arcctg(ctgx) =x; x2(0;):

49. arcsinx= arccosp

1x2= arctgx

p

1x2= arcctgp

1x2 x ;0< x <1:

50. arccosx= arcsinp

1x2= arctgp

1x2 x = arcctgx p

1x2;0< x <1:

51. arctgx= arcsinx

p

1 +x2= arccos1

p

1 +x2= arcctg1

x ;0< x <+1:

52. arcctgx= arcsin1

p

1 +x2= arccosx

p

1 +x2= arctg1

x ;0< x <+1:

53. arcsinx+arcsiny=2

6

664arcsin(xp

1y2+yp

1x2);dacaxy0 saux2+y21;

arcsin(xp

1y2+yp

1x2);dacax >0; y >0 six2+y2>1;

arcsin(xp

1y2+yp

1x2);dacax <0;y <0 six2+y2>1:

0

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

3

54. arcsinxarcsiny=2

6

664arcsin(xp

1y2yp

1x2);dacaxy0 saux2+y21;

arcsin(xp 1y2yp

1x2);dacax >0;y <0 six2+y2>1;

arcsin(xp 1y2yp

1x2);dacax <0;y >0 six2+y2>1:

55. arccosx+ arccosy=2

6

4arccos(xyq

(1x2)(1y2));dacax+y0;

2arccos(xyq

(1x2)(1y2));dacax+y <0:

56. arccosxarccosy=2

6

4arccos(xy+q

(1x2)(1y2));dacaxy; arccos(xy+q (1x2)(1y2));dacax < y:

57. arctgx+ arctgy=2

6

66666664arctg

x+y

1xy;dacaxy <1;

+ arctgx+y

1xy;dacax >0 sixy >1;

+ arctgx+y

1xy;dacax <0 sixy >1:

58. arctgxarctgy=2

6

66666664arctg

xy

1 +xy;dacaxy >1;

+ arctgxy

1 +xy;dacax >0 sixy <1;

+ arctgxy

1 +xy;dacax <0 sixy <1:

59. 2arcsinx=2

6

666666664arcsin(2xp

1x2);dacajxj p

2 2 arcsin(2xp

1x2);dacap

2 2 < x1; arcsin(2xp

1x2);daca1x 2 2

60. 2arccosx=2

4arccos(2x21) cand 0x1;

2arccos(2x21) cand1x <0:

61. 2arctgx=2

6

66666664arctg

2x

1x2;dacajxj<1;

+ arctg2x

1x2;dacax >1;

+ arctg2x

1x2;dacax <1:

0

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

4 62.
1 2 arcsinx=2 6

66664arcsins

1p 1x2 2 ;daca 0x1; arcsins 1p 1x2 2 ;daca1x <0: 63.
1 2 arccosx= arccoss 1 +x 2 ;daca1x1: 64.
1 2 arctgx=2 6

64arctgp

1 +x21

x ;dacax6= 0;

0;dacax= 0:

0

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

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