Преобразование тригонометрических выражений

Найти значение выражения

1.B tg²(arccos(- [pic] ))

2.B tg²(5arctg [pic] -0,25arcsin [pic] )

3.B [pic] tg(-arcsin(- [pic] ))

4.B log(-arccos(- [pic] )-arctg(- [pic] ))

5.B 5 [pic] sin( [pic] -arctg(- [pic] ))

6.B 6sin( [pic] +arcsin(- [pic] ))

7.B 10cos(arctg [pic] )

8.B [pic] arcsin(tg [pic] )

9.B arccos(tg [pic] )

10.B 3tg²(arcsin( [pic] ))

11.B tg(arcctg [pic] +6arcsin [pic] )

12B sin(arccos0,7+2arctg1)

13.B cos(arcsin0,3-6arccos [pic] )

14.B ctg(arctg3,5+5arcsin(-1))

15.B sin(14arcctg(-1)+2arccos(-0,35))

Упростить (найти значение выражения)

16. 2sin²x+3+2cos²x

1) 4 2) 2 3) 5 4) 6

17. 2-sin²( [pic] )-cos²( [pic] )

1) 3 2) 1 3) 2 4) 0

18. 8-5cos²x-5sin²x

1) 7 2) 13 3) -2 4) 3

19. 7sin²x+2+7cos²x

1) 16 2) 3 3) 9 4) 1

20. 9sin²y-6+9cos²y

1) 3 2) 4 3) 7 4) -15

21. 4sin [pic] cos [pic] при x=- [pic]

1) 0 2) 2 3) -1 4) –2

Упростить (найти значение выражения)

22. 8sin² [pic] -8cos² [pic] при x= [pic]

1) -4 [pic] 2) -4 3) 4 4) 4 [pic]

23. (sin2x-cos2x)² при x=- [pic]

1) - [pic] 2) 1 3) 1+ [pic] 4) 2

24. sin( [pic] )-sin( [pic] ) при x= [pic]

1) - [pic] 2) 0 3) [pic] 4) 1

25. [pic]

1) [pic] 2) 0,5 3) [pic] 4) [pic]

26. cos²x- [pic] при x=- [pic]

1) –0,25 2) 0,25 3) [pic] 4) [pic]

27. 1+cos(x+ [pic] )+cos(x- [pic] ) при x= [pic]

1) - [pic] 2) [pic] 3) 1 4) [pic]

28. 3-6sin² [pic] при x= [pic]

1) [pic] 2) [pic] 3) [pic] 4) [pic]

29. 4sin [pic] cos [pic] cos [pic] при x= [pic]

1) 0 2) 4 3) 1 4) [pic]

30. sin [pic] cos [pic] cos [pic] cos [pic] при x=- [pic]

1) 0 2) - [pic] 3) - [pic] 4) [pic]

31. [pic]

1) 1 2) tg² [pic] 3) ctg² [pic] 4) [pic]

32. (cosx-sinx)²+2sinxcosx

1) 1 2) 2 3) 1-2sin2x 4) cos2x+sin2x

33. tg [pic] -2sin(- [pic] )-cos3

1) [pic] 2) 1 3) [pic] -2 4) - [pic]

Упростить (найти значение выражения)

34. [pic]

1) [pic] 2) [pic] 3) [pic] 4) [pic]

35. cos( [pic] )-2sin [pic] sin [pic] , если [pic] =42^о, [pic] =18^o

1) –0,5 2) [pic] 3) 0,5 4) [pic]

36. [pic]

1) cos⁴ [pic] 2) cos2 [pic] 3) tg² [pic] -sin² [pic] 4) cos² [pic]

37. cosx+tgxsinx

1) 1 2) 2cosx 3) cosx+sinx 4) [pic]

38. [pic] -tg² [pic]

1) ctg2 [pic] 2) 0 3) ctg² [pic] -tg² [pic] 4) 2tg² [pic]

39. 1-sin [pic] ctg [pic] cos [pic]

1) 0 2) sin² [pic] 3) cos² [pic] 4) 1-sin2 [pic]

40. cos-sin(- [pic] )+tg² [pic]

1) [pic] 2) 3 3) [pic] -2 4) 1

41. [pic]

1) 3 [pic] 2) 3 3) 1,5 [pic] 4) [pic]

42. [pic]

1) -1 2) [pic] 3)- [pic] 4) 1

43. 1+ctg( [pic] +x)sinxcosx

1) sin²x 2) 1+sin²x 3) cos²x 4) 1+cos²x

44. 1- [pic]

1) 2cos³ [pic] 2) sin² [pic] 3) cos [pic] 4) cos² [pic]

45. cos²( [pic] )+cos²( [pic] )

1) 1 2) 2cos² [pic] 3) 2sin² [pic] 4) 0

46. [pic]

1) 1 2) [pic] 3) [pic] 4) 1+sin [pic]

47. [pic]

1) 1 2) [pic] 3) [pic] 4) [pic]

Упростить (найти значение выражения)

48. log [pic] (4cos [pic] )+log [pic] (sin [pic] )

1) [pic] 2) [pic] 3) 1 4) 4

49. cos2x+tg²xcos²x

1) 1 2) sin²x 3) sin2x 4) cos²x

50. sin⁴ [pic] -cos4 [pic] +cos² [pic]

1) 1 2) sin² [pic] +2cos² [pic] 3) sin² [pic] 4) 0

51. [pic]

1) ctg² [pic] -ctg [pic] 2) [pic] 3) [pic] 4) 1+cos [pic]

52. [pic]

1) tg² [pic] -tg [pic] 2) 1+sin [pic] 3) [pic] 4) [pic]

53. [pic]

1) tg⁴ [pic] 2) tg² [pic] +1 3) sin² [pic] 4) cos² [pic]

54. [pic]

1) [pic] 2) cos² [pic] 3) tg² [pic] 4) 1

55. (sin [pic] +cos [pic] )²+( sin [pic] -cos [pic] )

1) -2sin2 [pic] 2) 2 3) 2sin² [pic] 4) 4

56. -sin² [pic] -cos² [pic] -tg² [pic]

1) -sin² [pic] 2) –cos2 [pic] -tg² [pic] 3) - [pic] 4) –cos² [pic]

57. [pic]

1) sin [pic] 2) tg² [pic] 3) ctg² [pic] 4) [pic]

Упростить

58.B (tg [pic] +ctg [pic] )²-(tg [pic] -ctg [pic] )²

59.B [pic]

60.B [pic]

61.B ctg² [pic] -cos² [pic] -ctg² [pic] cos² [pic]

62.B [pic]

63B [pic]

64B ((1-tg [pic] )²+(1+tg [pic] )²) cos² [pic]

Упростить

65.B [pic]

66.B sin [pic] tg [pic] - [pic] при [pic] =- [pic]

67.B [pic] при [pic] = [pic]

68.B 2(cos8^ocos37^o-cos82^ocos53^o)²

69.B [pic] при [pic] = [pic] , [pic] = [pic]

70.B sin²(26^o+ [pic] )+sin²(244^o- [pic] )+tg(113^o+ [pic] )ctg(67^o- [pic] ) при [pic] =14^o

71.B [pic]

72.B [pic]

73.B [pic]

74.B [pic]

75.B [pic] [pic]

76.B (- [pic] +2)tg105^o

77.B 4(sin [pic] +sin [pic] -cos [pic] )²

78.B [pic]

79.B tg [pic] +ctg [pic]

80.B [pic]

81.B 2sin15^osin75^o

82.B 4(sin [pic] +sin2 [pic] +sin3 [pic] +sin4 [pic] )² при [pic] = [pic]

83.B [pic] при [pic] = [pic]

84.B 4( [pic] при x= [pic]

85.B [pic] при [pic] = [pic]

86.B [pic] при [pic] = [pic]

Упростить

87.B [pic] при [pic] = [pic]

88.B [pic] при [pic] = [pic]

89.B [pic] при [pic] = [pic]

90.B [pic] при [pic] = [pic]

91.B cos(2-3x)cosx+sin3xcos( [pic] +x) при x= [pic]

92. 1-2sin^{2 [pic]}

1) [pic] 2) [pic] 3) [pic] 4) 1

93. [pic]

1) 1 2) 2 3) [pic] 4) -4

94. [pic]

1) 2 2) 1 3) 2 [pic] 4) 4

95. (1+tg²( [pic] ))cos² [pic]

1) ctg² [pic] 2) tg² [pic] 3) 1 4) cos⁴ [pic]

96. sin17^ocos28^o+cos17^osin28^o

1) - [pic] 2) [pic] 3) [pic] 4) - [pic]

97. -cos57^ocos33^o+sin57^osin33^o

1) -1 2) [pic] 3) - [pic] 4) 0

98. [pic]

1) - [pic] 2) [pic] 3) [pic] 4) 60

99. [pic]

1) - [pic] 2) - [pic] 3) [pic] 4) [pic]

100. -cos² [pic] +sin² [pic]

1) - [pic] 2) [pic] 3) - [pic] 4) [pic]

101. 2cos² [pic] -1

1) 1 2) [pic] 3) - [pic] 4) 0

102. [pic]

1) [pic] 2) 1 3) - [pic] 4) -1

103. [pic]

1) [pic] 2) - [pic] 3) [pic] 4) - [pic]

104. cos⁴ [pic] -sin⁴ [pic]

1) - [pic] 2) - [pic] 3) [pic] 4) [pic]

105. [pic]

1) - [pic] 2) [pic] 3) [pic] 4) - [pic]

106. cos( [pic] )-0,5sin [pic]

1) [pic] cos [pic] 2) [pic] cos [pic] 3) cos [pic] -sin [pic] 4) cos [pic] +sin [pic]

107. [pic]

1) cos [pic] -sin [pic] 2) ctg2 [pic] 3) [pic] 4) tg [pic]

108. [pic]

1) tg [pic] 2) tg2 [pic] 3) [pic] 4) 1+cos [pic]

109. [pic]

1) - [pic] 2) - [pic] 3) [pic] 4) [pic]

110. [pic]

1) 0 2) tg [pic] -ctg [pic] 3) sin² [pic] 4) cos² [pic]

111. [pic]

1) ctg² [pic] 2) tg² [pic] 3) sin² [pic] 4) cos² [pic]

112. [pic]

1) 1 2) -ctg² [pic] 3) -1 4) tg² [pic]

113. [pic]

1) 1 2) [pic] 3) [pic] 4) -1

114. [pic]

1) -sin [pic] cos [pic] 2) cos² [pic] 3) sin² [pic] 4) sin [pic] cos [pic]

115. [pic]

1) ctg [pic] 2) tg [pic] 3) –tg² [pic] 4) –ctg² [pic]

116. sin2,5 [pic] cos1,5 [pic] +sin1,5 [pic] cos2,5 [pic] +cos(4- [pic] )

1) sin4 [pic] -cos [pic] 2) sin [pic] +cos [pic]

3) sin [pic] -cos [pic] 4) cos [pic] +sin4 [pic]

117. (tg [pic] +ctg [pic] )²-2 при [pic] =- [pic]

1) -2 2) 2 3) -1 4) 0

118. [pic]

1) 1 2) [pic] 3) [pic] 4) 1+sin [pic]

119. tg²(270^o+ [pic] )sin²(180^o+ [pic] )

1) -sin² [pic] 2) cos² [pic] 3) sin² [pic] 4) -cos² [pic]

120.B cos15^o(cos35^osin50^o-cos50^osin35^o)

121.B cos45^o(cos25^osin70^o-cos70^osin25^o)

122.B cos15^o(sin5^ocos10^o+cos5^osin10^o)

123.B sin15^o(cos116^ocos101^o+sin116^osin101^o)

124.B 4sin10^osin20^o-2cos10^o

125.B cos100^ocos20^o-0,5cos80^o

Тригонометрические уравнения

Укажите наименьший положительный корень уравнения

1. sin(35^o+x)= [pic]

1) 5^o 2) 110^o 3) 15^o 4) 10^o

2. tg(3x+45^o)= [pic]

1) 5^o 2) 55^o 3) 165^o 4) 45^o

3. tg(3x+30^o)= [pic]

1) 5^o 2) 30^o 3) 75^o 4) 10^o

Решите уравнение

4. cos²x-sin²x=- [pic]

1) n, n [pic] Z 2) [pic] e, e [pic] Z

3) [pic] , k [pic] Z 4) [pic] , m [pic] Z

5. sin(-x)-cos( [pic] +x)= [pic]

1) (-1)ⁿ [pic] +n, n [pic] Z 2) (-1)^k [pic] +k, k [pic] Z

3) [pic] +2e, e [pic] Z 4) [pic] +2m, m [pic] Z

6. cos(+x)=sin [pic]

1) [pic] +m, m [pic] Z 2) 2e, e [pic] Z

3) +2n, n [pic] Z 4) [pic] +k, k [pic] Z

7. 2sinxcosx= [pic]

1) [pic] +k, k [pic] Z 2) (-1)ⁿ [pic] + [pic] , n [pic] Z

3) (-1)^k [pic] +k, k [pic] Z 4) [pic] +2e, e [pic] Z

8. 3cosx-sin2x=0

1) [pic] +2n, n [pic] Z 2) 2n, n [pic] Z

3) [pic] + [pic] n, n [pic] Z 4) [pic] +n, n [pic] Z

9. 4sinx+sin2x=0

1) корней нет 2) 2n, n [pic] Z 3) n, n [pic] Z 4) [pic] +n, n [pic] Z

Решите уравнение

10. 3sinx=sin2x

1) + [pic] n, n [pic] Z 2) [pic] n, n [pic] Z 3) n, n [pic] Z 4) корней нет

11. cos2-sin²x=0,5

1) [pic] +n, n [pic] Z 2) [pic] +2n, n [pic] Z

3) [pic] +2n, n [pic] Z 4) [pic] +n, n [pic] Z

11. cos2-sin²x=0,5

1) [pic] +n, n [pic] Z 2) [pic] +2n, n [pic] Z

3) [pic] +2n, n [pic] Z 4) [pic] +n, n [pic] Z

12. 9sin4x=0

1) [pic] +n, n [pic] Z 2) [pic] n, n [pic] Z

3) [pic] +2n, n [pic] Z 4) +n, n [pic] Z

13. 1+tg²x- [pic] =sinx- [pic]

1) [pic] +2n, n [pic] Z 2) (-1)ⁿ [pic] +n, n [pic] Z

3) [pic] + [pic] n, n [pic] Z 4) [pic] +2n, n [pic] Z

14. 1+tg²x=sinx-0,5+ [pic]

1) n, n [pic] Z 2) [pic] +n, n [pic] Z

3) (-1)ⁿ [pic] +n, n [pic] Z 4) (-1)ⁿ [pic] +n, n [pic] Z

15. [pic] =cos²x+cosx-1

1) 2n, n [pic] Z 2) [pic] +2n, n [pic] Z

3) n, n [pic] Z 4) [pic] +n, n [pic] Z

16. 2cosx+1= [pic] +1-ctg²x

1) [pic] n, n [pic] Z 2) (-1)ⁿ [pic] +n, n [pic] Z

3) [pic] +2n, n [pic] Z 4) 2n, n [pic] Z

17. ctg²x(1-cos²x)=0

1) [pic] +n, n [pic] Z 2) [pic] n, n [pic] Z

3) n, n [pic] Z 4) [pic] +n, n [pic] Z

18. 2sinx(tg²x+1)= [pic]

1) n, n [pic] Z 2) (-1)ⁿ [pic] +n, n [pic] Z

3) [pic] +n, n [pic] Z 4) [pic] +2n, n [pic] Z

19. cosx+ctg²x=- [pic] + [pic]

1) (-1)ⁿ [pic] +n, n [pic] Z 2) [pic] +2n, n [pic] Z

3) n, n [pic] Z 4) [pic] +2n, n [pic] Z

20. sin4xcos2x-cos4xsin2x=0

1) [pic] n, n [pic] Z 2) (-1)ⁿ [pic] +n, n [pic] Z

3) [pic] +n, n [pic] Z 4) (-1)ⁿ [pic] +2n, n [pic] Z

21. sin [pic] xcos [pic] x-cos [pic] xsin [pic] x=- [pic]

1) (-1)ⁿ⁺¹ [pic] +n, n [pic] Z 2) (-1)ⁿ⁺¹ [pic] +n, n [pic] Z

3) (-1)ⁿ [pic] +n, n [pic] Z 4) (-1)ⁿ⁺¹ [pic] + [pic] n, n [pic] Z

22. [pic] =(1+ctg²x)sinx=0

1) - [pic] +2n, n [pic] Z 2) [pic] +2n, n [pic] Z

3) [pic] +n, n [pic] Z 4) 2n, n [pic] Z

23. Укажите число корней на промежутке (- [pic] ;)

tg²x-tgx=0

1) 1 2) 2 3) 3 4) 4

Определите число корней на промежутке

24. ctg5xsin10x-cos10x-cos20x=0 [0,2]

25. tg4xsin8x+cos8x-cos16x=0 [0,2]

26. tg6xsin12x+cos12x-cos24x=0 [0,2]

27. tgxcos5x+sin5x=sin6x [ [pic] , [pic] ]

28. tg2xcos6x-sin6x=sin4x [ [pic] , [pic] ]

29. tg3xcosx-sinx=sin2x [ [pic] , [pic] ]

30. tg3xsin6x+cos6x-cos12x=0 [0,2]

31. cos4x+cos2x-ctgxsin2x=0 [0,2]

Найти сумму корней уравнения на отрезке

32. 4sin²7x*cos²7x+sin²( [pic] +14x)= [pic] +cos( [pic] - [pic] ) на отрезке [3,5]

33. 4sin²5x*cos²5x+sin²( [pic] -10x)= [pic] +cos [pic]

на отрезке [-1,3]

34. [pic] +cosx=tg²x

1) [pic] +2n, n [pic] Z 2) 2n, n [pic] Z

3) n, n [pic] Z 4) +2n, n [pic] Z

35. (tg²x+1)tgx=- [pic]

1) - [pic] +2n, n [pic] Z 2) n, n [pic] Z

3) - [pic] +n, n [pic] Z 4) [pic] +n, n [pic] Z

36. cos(3x- [pic] )=- [pic]

1) - [pic] [pic] + [pic] n, n [pic] Z 2) [pic] [pic] + [pic] n, n [pic] Z

3) [pic] [pic] + [pic] n, n [pic] Z 4) [pic] [pic] + [pic] n, n [pic] Z

37. 2sin(x+ [pic] )=1

1) - [pic] +(-1)ⁿ [pic] +n, n [pic] Z 2) [pic] -(-1)ⁿ [pic] +n, n [pic] Z

3) (-1)ⁿ [pic] - [pic] +n, n [pic] Z 4) (-1)ⁿ [pic] + [pic] +2n, n [pic] Z

38. 2sin(2x- [pic] )= [pic]

1) (-1)ⁿ [pic] + [pic] +n, n [pic] Z 2) [pic] +(-1)ⁿ [pic] + [pic] n, n [pic] Z

3) [pic] [pic] +n, n [pic] Z 4) (-1)ⁿ [pic] - [pic] +n, n [pic] Z

39. cos( [pic] -2x)= [pic]

1) [pic] +2n, n [pic] Z 2) [pic] +n, n [pic] Z

3) (-1)ⁿ [pic] +n, n [pic] Z 4) (-1)ⁿ [pic] + [pic] n, n [pic] Z

40. 2tg(3x-)=0

1) 2n, n [pic] Z 2) [pic] n, n [pic] Z

3) 3n, n [pic] Z 4) [pic] + [pic] n, n [pic] Z

41. 2sin( [pic] -x)=-1

1) [pic] +2n, n [pic] Z 2) (-1)ⁿ [pic] +n, n [pic] Z

3) [pic] +2n, n [pic] Z 4) (-1)ⁿ [pic] +n, n [pic] Z

42. 2cos(+2x)= [pic]

1) (-1)ⁿ [pic] +n, n [pic] Z 2) [pic] +n, n [pic] Z

3) [pic] +2n, n [pic] Z 4) (-1)ⁿ [pic] + [pic] n, n [pic] Z

43. sinx+cosx=0

1) - [pic] +n, n [pic] Z 2) - [pic] + [pic] n, n [pic] Z

3) [pic] +n, n [pic] Z 4) - [pic] +2n, n [pic] Z

44. sinx [pic] cosx=0

1) [pic] +n, n [pic] Z 2) - [pic] +2n, n [pic] Z

3) [pic] +n, n [pic] Z 4) [pic] +2n, n [pic] Z

45. sin( [pic] +x)= [pic]

1) - [pic] +k, k [pic] Z 2) [pic] +2k, k [pic] Z

3) [pic] +2k, k [pic] Z 4) (-1)^k [pic] +k, k [pic] Z

46. cos( [pic] -x)= [pic]

1) (-1)ⁿ [pic] +n, n [pic] Z 2) (-1)ⁿ [pic] +n, n [pic] Z

3) [pic] +2n, n [pic] Z 4) [pic] +2k, k [pic] Z

47. sinx= [pic] cosx

1) [pic] +n, n [pic] Z 2) [pic] +2n, n [pic] Z

3) [pic] +2n, n [pic] Z 4) n, n [pic] Z

48. cos²x=sin²x

1) [pic] +2k, k [pic] Z 2) [pic] + [pic] k, k [pic] Z

3) [pic] +2k, k [pic] Z 4) [pic] +k, k [pic] Z

49. tg(x- [pic] )=-1

1) [pic] +n, n [pic] Z 2) [pic] +n, n [pic] Z

3) n, n [pic] Z 4) [pic] +n, n [pic] Z

50. sinx=tg [pic]

1) (-1)ⁿ [pic] +n, n [pic] Z 2) (-1)ⁿ [pic] +n, n [pic] Z

3) n, n [pic] Z 4) нет корней

51. sin²x-2sinx=0

1) [pic] +n, n [pic] Z 2) - [pic] +n, n [pic] Z

3) n, n [pic] Z 4) 2n, n [pic] Z

52. 8cos²x-2cosx-1=0

53. cos( [pic] - [pic] )= [pic]

54. (2sinx-3)(cosx+1)=0

55. ctg3x=-1

56. sin( [pic] +x)+3cosx=0

57. cos²x-sin²x=cos

58. Найти произведение корней уравнения

sin2x=x

1) - [pic] 2) - [pic] 3) 0 4) корней нет

Первообразные и площади фигур

Найти общий вид первообразных

1. f(x)=2sinx

1) 2cosx+C 2) 2cosx+C 3) - [pic] +C 4) -2sinx+C

2. f(x)= [pic] +2

1) tgx+C 2) tgx+2x+C 3) - [pic] +C 4) ctgx-2x+C

3. f(x)= [pic]

4. f(x)=-sinx-cosx

5. f(x)= [pic] -cosx

6. f(x)=cos( [pic] -x)

Найдите значение первообразной функции при x0, график которой проходит через точку M

7. f(x)= [pic] x0= [pic] ; M( [pic] ;- [pic] )

8. f(x)= [pic] x0= [pic] ; M(0;1 [pic] )

9. f(x)=-2cos( [pic] -x) x0= [pic] ; M(- [pic] ;2)

10. f(x)=- [pic] cos( [pic] - [pic] ) x0=-2, [pic] ; M(;1,5)

Найти площадь фигуры, ограниченной линиями

11. f(x)= [pic] x= [pic] ; x= [pic]

12. y=cosx x=- [pic] ; x= [pic]

13. y=- [pic] x= [pic] ; x= [pic]

14. y=4x-x² x=1; x=3; y=0

15. y= [pic] x=e; x=e²; y=0

16. y=x³ x=1; x=2; y=0

17. y=x⁴ x=1; x=2; y=0

18. y=x⁴+x³ x=1; x=2; y=0

19. y=x³-x² x=2; x=3; y=0

20. y=3x²-2x+1 y=0; x=0; x=3

21. y=e^x-e^-x x=ln2; x=ln3; y=0

Найти площадь фигуры, ограниченной линиями

22. [pic] Sф-? y=-x²+4x-3; y=0

23. 3Sф-? y= [pic] ; y=0; x=4

24. y=2-sinx y=2; x=0; x=

25. y=x³; y=8; x=0

26. y=(x-2)²; y=4; x=1; x=4

27. y=(x-1)²; y=-x+3