Keynote de la semana 11 de mi curso de trigonometría.

1. Day 51
π
1. Quiz 6 Feedback 2
y π
2π ⎛ − 1 , 3 ⎞ (0, 1)
⎛ 1 , 3 ⎞
3 ⎝ 2 2 ⎠ ⎝ 2 2 ⎠ 3
3π ⎛ − 2 , 2 ⎞ ⎛ 2 , 2 ⎞ π
4 ⎝ 2 2 ⎠ ⎝ 2 2 ⎠ 4
5π ⎛ − 3 , 1 ⎞ ⎛ 3 , 1 ⎞ π
6 ⎝ 2 2 ⎠ ⎝ 2 2 ⎠ 6
(-1, 0) (1, 0)
π 2π
x
7π ⎛ − 3 , − 1 ⎞ ⎛ 3 , − 1 ⎞ 11π
6 ⎝ 2 2 ⎠ ⎝ 2 2 ⎠ 6
5π ⎛ − 2 , − 2 ⎞ ⎛ 2 , − 2 ⎞ 7π
4 ⎝ 2 2 ⎠ ⎝ 2 2 ⎠ 4
4π ⎛ − 1 2 , − 3 2 ⎞ ⎛ 1 , − 3 ⎞ 5π
3 ⎝ ⎠ 3π (0, -1) ⎝ 2 2 ⎠ 3
2

2. 2. Examples
Sketch the graph of y = −3cos ( x + π ) + 1

3. 2. Examples
Sketch the graph of y = −3cos ( x + π ) + 1

4. 2. Examples
Sketch the graph of y = −3cos ( x + π ) + 1
y = cos ( x )

5. 2. Examples
Sketch the graph of y = −3cos ( x + π ) + 1
y = −3cos ( x )

6. 2. Examples
Sketch the graph of y = −3cos ( x + π ) + 1
y = −3cos ( x + π )

7. 2. Examples
Sketch the graph of y = −3cos ( x + π ) + 1
y = −3cos ( x + π ) + 1

8. 2. Examples
Sketch the graph of ( )
y = 2sin x − π 3 − 1

9. 2. Examples
Sketch the graph of ( )
y = 2sin x − π 3 − 1

10. 2. Examples
Sketch the graph of ( )
y = 2sin x − π 3 − 1
y = sin ( x )

11. 2. Examples
Sketch the graph of ( )
y = 2sin x − π 3 − 1
y = 2sin ( x )

12. 2. Examples
Sketch the graph of ( )
y = 2sin x − π 3 − 1
(
y = 2sin x − π 3 )

13. 2. Examples
Sketch the graph of ( )
y = 2sin x − π 3 − 1
( )
y = 2sin x − π 3 − 1

14. 2. Examples
Sketch the graph of y = 3cos ( 2x + π ) + 1

15. 2. Examples
Sketch the graph of y = 3cos ( 2x + π ) + 1

16. 2. Examples
Sketch the graph of ( )
y = 2sin 2x − π 3 − 1

17. 2. Examples
Sketch the graph of ( )
y = 2sin 2x − π 3 − 1

18. Day 52
1. Examples
Find the amplitude and period of y = 3 sin 4x and sketch the graph of
the function.

19. Day 52
1. Examples
Find the amplitude and period of y = 3 sin 4x and sketch the graph of
the function.
If , the amplitude and period of
and are given by:

20. Day 52
1. Examples
Find the amplitude and period of y = 3 sin 4x and sketch the graph of
the function.
If ω > 0 , the amplitude and period of
and are given by:

21. Day 52
1. Examples
Find the amplitude and period of y = 3 sin 4x and sketch the graph of
the function.
If ω > 0 , the amplitude and period of y = Asin ω x
and are given by:

22. Day 52
1. Examples
Find the amplitude and period of y = 3 sin 4x and sketch the graph of
the function.
If ω > 0 , the amplitude and period of y = Asin ω x
and y = A cos ω x are given by:

23. Day 52
1. Examples
Find the amplitude and period of y = 3 sin 4x and sketch the graph of
the function.
If ω > 0 , the amplitude and period of y = Asin ω x
and y = A cos ω x are given by:
Amplitude =

24. Day 52
1. Examples
Find the amplitude and period of y = 3 sin 4x and sketch the graph of
the function.
If ω > 0 , the amplitude and period of y = Asin ω x
and y = A cos ω x are given by:
Amplitude = A

25. Day 52
1. Examples
Find the amplitude and period of y = 3 sin 4x and sketch the graph of
the function.
If ω > 0 , the amplitude and period of y = Asin ω x
and y = A cos ω x are given by:
Amplitude = A Period = T =

26. Day 52
1. Examples
Find the amplitude and period of y = 3 sin 4x and sketch the graph of
the function.
If ω > 0 , the amplitude and period of y = Asin ω x
and y = A cos ω x are given by:
2π
Amplitude = A Period = T =
ω

27. 1. Examples
Find the amplitude and period of y = 3 sin 4x and sketch the graph of
the function.

28. 2. Phase Shift
Find the amplitude and period of y = 3sin ( 2x − π ) and sketch the
graph of the function.

29. 2. Phase Shift
Find the amplitude and period of y = 3sin ( 2x − π ) and sketch the
graph of the function.
If , for the graphs of
and are given by:

30. 2. Phase Shift
Find the amplitude and period of y = 3sin ( 2x − π ) and sketch the
graph of the function.
If ω > 0 , for the graphs of
and are given by:

31. 2. Phase Shift
Find the amplitude and period of y = 3sin ( 2x − π ) and sketch the
graph of the function.
If ω > 0 , for the graphs of y = Asin (ω x − φ )
and are given by:

32. 2. Phase Shift
Find the amplitude and period of y = 3sin ( 2x − π ) and sketch the
graph of the function.
If ω y = Asin (ω x − φ )
> 0 , for the graphs of
and y = A cos (ω x − φ ) are given by:

33. 2. Phase Shift
Find the amplitude and period of y = 3sin ( 2x − π ) and sketch the
graph of the function.
If ω y = Asin (ω x − φ )
> 0 , for the graphs of
and y = A cos (ω x − φ ) are given by:
Amplitude =

34. 2. Phase Shift
Find the amplitude and period of y = 3sin ( 2x − π ) and sketch the
graph of the function.
If ω y = Asin (ω x − φ )
> 0 , for the graphs of
and y = A cos (ω x − φ ) are given by:
Amplitude = A

35. 2. Phase Shift
Find the amplitude and period of y = 3sin ( 2x − π ) and sketch the
graph of the function.
If ω y = Asin (ω x − φ )
> 0 , for the graphs of
and y = A cos (ω x − φ ) are given by:
Amplitude = A Period = T =

36. 2. Phase Shift
Find the amplitude and period of y = 3sin ( 2x − π ) and sketch the
graph of the function.
If ω y = Asin (ω x − φ )
> 0 , for the graphs of
and y = A cos (ω x − φ ) are given by:
2π
Amplitude = A Period = T =
ω

37. 2. Phase Shift
Find the amplitude and period of y = 3sin ( 2x − π ) and sketch the
graph of the function.
If ω y = Asin (ω x − φ )
> 0 , for the graphs of
and y = A cos (ω x − φ ) are given by:
2π
Amplitude = A Period = T = Phase shift =
ω

38. 2. Phase Shift
Find the amplitude and period of y = 3sin ( 2x − π ) and sketch the
graph of the function.
If ω y = Asin (ω x − φ )
> 0 , for the graphs of
and y = A cos (ω x − φ ) are given by:
2π φ
Amplitude = A Period = T = Phase shift =
ω ω

39. 2. Phase Shift
Find the amplitude and period of y = 3sin ( 2x − π ) and sketch the
graph of the function.

40. 3. Classwork
Solve the given exercises.

41. 4. Answers
( )
y = 3cos x − π 3 + 1

42. 4. Answers
( )
y = −2sin x − π 2 − 1

43. 4. Answers
( )
y = 3cos 2x − π 3 + 1

44. 4. Answers
( )
y = −2sin 2x − π 3 − 1

45. Day 53
1. Homework check
y = sin x

46. 1. Homework check
y = cos x

47. 2. Graphs of Trigonometric Functions

48. 2. Graphs of Trigonometric Functions

49. 2. Graphs of Trigonometric Functions
y = sin x

50. 2. Graphs of Trigonometric Functions
y = sin x
y = cos x

51. 2. Graphs of Trigonometric Functions
y = sin x
y = cos x
sin x
y = tan x =
cos x

52. 2. Graphs of Trigonometric Functions
y = sin x
y = cos x
sin x
y = tan x =
cos x

53. 2. Graphs of Trigonometric Functions
y = sin x
y = cos x
sin x
y = tan x =
cos x
y = tan x

54. 2. Graphs of Trigonometric Functions

55. 2. Graphs of Trigonometric Functions

56. 2. Graphs of Trigonometric Functions
y = sin x

57. 2. Graphs of Trigonometric Functions
y = sin x
y = cos x

58. 2. Graphs of Trigonometric Functions
y = sin x
y = cos x
cos x
y = cot x =
sin x

59. 2. Graphs of Trigonometric Functions
y = sin x
y = cos x
cos x
y = cot x =
sin x
y = cot x

60. 2. Graphs of Trigonometric Functions
y = sin x
y = cos x
cos x
y = cot x =
sin x
y = cot x

61. 2. Graphs of Trigonometric Functions

62. 2. Graphs of Trigonometric Functions

63. 2. Graphs of Trigonometric Functions
y = sin x

64. 2. Graphs of Trigonometric Functions
y = sin x
y = cos x

65. 2. Graphs of Trigonometric Functions
y = sin x
y = cos x
1
y = sec x =
cos x

66. 2. Graphs of Trigonometric Functions
y = sin x
y = cos x
1
y = sec x =
cos x
y = sec x

67. 2. Graphs of Trigonometric Functions
y = sin x
y = cos x
1
y = sec x =
cos x
y = sec x

68. 2. Graphs of Trigonometric Functions

69. 2. Graphs of Trigonometric Functions

70. 2. Graphs of Trigonometric Functions
y = sin x

71. 2. Graphs of Trigonometric Functions
y = sin x
y = cos x

72. 2. Graphs of Trigonometric Functions
y = sin x
y = cos x
1
y = csc x =
sin x

73. 2. Graphs of Trigonometric Functions
y = sin x
y = cos x
1
y = csc x =
sin x
y = csc x

74. 2. Graphs of Trigonometric Functions
y = sin x
y = cos x
1
y = csc x =
sin x
y = csc x