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# Week 11 - Trigonometry

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Keynote de la semana 11 de mi curso de trigonometría.

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### Week 11 - Trigonometry

1. 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 ⎠ 45π ⎛ − 3 , 1 ⎞ ⎛ 3 , 1 ⎞ π 6 ⎝ 2 2 ⎠ ⎝ 2 2 ⎠ 6 (-1, 0) (1, 0)π 2π x7π ⎛ − 3 , − 1 ⎞ ⎛ 3 , − 1 ⎞ 11π 6 ⎝ 2 2 ⎠ ⎝ 2 2 ⎠ 65π ⎛ − 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. 2. ExamplesSketch the graph of y = −3cos ( x + π ) + 1
3. 3. 2. ExamplesSketch the graph of y = −3cos ( x + π ) + 1
4. 4. 2. ExamplesSketch the graph of y = −3cos ( x + π ) + 1 y = cos ( x )
5. 5. 2. ExamplesSketch the graph of y = −3cos ( x + π ) + 1 y = −3cos ( x )
6. 6. 2. ExamplesSketch the graph of y = −3cos ( x + π ) + 1 y = −3cos ( x + π )
7. 7. 2. ExamplesSketch the graph of y = −3cos ( x + π ) + 1 y = −3cos ( x + π ) + 1
8. 8. 2. ExamplesSketch the graph of ( ) y = 2sin x − π 3 − 1
9. 9. 2. ExamplesSketch the graph of ( ) y = 2sin x − π 3 − 1
10. 10. 2. ExamplesSketch the graph of ( ) y = 2sin x − π 3 − 1 y = sin ( x )
11. 11. 2. ExamplesSketch the graph of ( ) y = 2sin x − π 3 − 1 y = 2sin ( x )
12. 12. 2. ExamplesSketch the graph of ( ) y = 2sin x − π 3 − 1 ( y = 2sin x − π 3 )
13. 13. 2. ExamplesSketch the graph of ( ) y = 2sin x − π 3 − 1 ( ) y = 2sin x − π 3 − 1
14. 14. 2. ExamplesSketch the graph of y = 3cos ( 2x + π ) + 1
15. 15. 2. ExamplesSketch the graph of y = 3cos ( 2x + π ) + 1
16. 16. 2. ExamplesSketch the graph of ( ) y = 2sin 2x − π 3 − 1
17. 17. 2. ExamplesSketch the graph of ( ) y = 2sin 2x − π 3 − 1
18. 18. Day 521. ExamplesFind the amplitude and period of y = 3 sin 4x and sketch the graph of the function.
19. 19. Day 521. ExamplesFind the amplitude and period of y = 3 sin 4x and sketch the graph of the function.If , the amplitude and period ofand are given by:
20. 20. Day 521. ExamplesFind the amplitude and period of y = 3 sin 4x and sketch the graph of the function.If ω > 0 , the amplitude and period ofand are given by:
21. 21. Day 521. ExamplesFind 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 xand are given by:
22. 22. Day 521. ExamplesFind 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 xand y = A cos ω x are given by:
23. 23. Day 521. ExamplesFind 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 xand y = A cos ω x are given by:Amplitude =
24. 24. Day 521. ExamplesFind 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 xand y = A cos ω x are given by:Amplitude = A
25. 25. Day 521. ExamplesFind 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 xand y = A cos ω x are given by:Amplitude = A Period = T =
26. 26. Day 521. ExamplesFind 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 xand y = A cos ω x are given by: 2πAmplitude = A Period = T = ω
27. 27. 1. ExamplesFind the amplitude and period of y = 3 sin 4x and sketch the graph of the function.
28. 28. 2. Phase ShiftFind the amplitude and period of y = 3sin ( 2x − π ) and sketch the graph of the function.
29. 29. 2. Phase ShiftFind the amplitude and period of y = 3sin ( 2x − π ) and sketch the graph of the function.If , for the graphs ofand are given by:
30. 30. 2. Phase ShiftFind the amplitude and period of y = 3sin ( 2x − π ) and sketch the graph of the function.If ω > 0 , for the graphs ofand are given by:
31. 31. 2. Phase ShiftFind 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. 32. 2. Phase ShiftFind 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 ofand y = A cos (ω x − φ ) are given by:
33. 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. 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. 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. 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: 2πAmplitude = A Period = T = ω
37. 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: 2πAmplitude = A Period = T = Phase shift = ω
38. 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 φAmplitude = A Period = T = Phase shift = ω ω
39. 39. 2. Phase ShiftFind the amplitude and period of y = 3sin ( 2x − π ) and sketch the graph of the function.
40. 40. 3. ClassworkSolve the given exercises.
41. 41. 4. Answers ( ) y = 3cos x − π 3 + 1
42. 42. 4. Answers ( ) y = −2sin x − π 2 − 1
43. 43. 4. Answers ( ) y = 3cos 2x − π 3 + 1
44. 44. 4. Answers ( ) y = −2sin 2x − π 3 − 1
45. 45. Day 531. Homework check y = sin x
46. 46. 1. Homework check y = cos x
47. 47. 2. Graphs of Trigonometric Functions
48. 48. 2. Graphs of Trigonometric Functions
49. 49. 2. Graphs of Trigonometric Functions y = sin x
50. 50. 2. Graphs of Trigonometric Functions y = sin x y = cos x
51. 51. 2. Graphs of Trigonometric Functions y = sin x y = cos x sin x y = tan x = cos x
52. 52. 2. Graphs of Trigonometric Functions y = sin x y = cos x sin x y = tan x = cos x
53. 53. 2. Graphs of Trigonometric Functions y = sin x y = cos x sin x y = tan x = cos x y = tan x
54. 54. 2. Graphs of Trigonometric Functions
55. 55. 2. Graphs of Trigonometric Functions
56. 56. 2. Graphs of Trigonometric Functions y = sin x
57. 57. 2. Graphs of Trigonometric Functions y = sin x y = cos x
58. 58. 2. Graphs of Trigonometric Functions y = sin x y = cos x cos x y = cot x = sin x
59. 59. 2. Graphs of Trigonometric Functions y = sin x y = cos x cos x y = cot x = sin x y = cot x
60. 60. 2. Graphs of Trigonometric Functions y = sin x y = cos x cos x y = cot x = sin x y = cot x
61. 61. 2. Graphs of Trigonometric Functions
62. 62. 2. Graphs of Trigonometric Functions
63. 63. 2. Graphs of Trigonometric Functions y = sin x
64. 64. 2. Graphs of Trigonometric Functions y = sin x y = cos x
65. 65. 2. Graphs of Trigonometric Functions y = sin x y = cos x 1 y = sec x = cos x
66. 66. 2. Graphs of Trigonometric Functions y = sin x y = cos x 1 y = sec x = cos x y = sec x
67. 67. 2. Graphs of Trigonometric Functions y = sin x y = cos x 1 y = sec x = cos x y = sec x
68. 68. 2. Graphs of Trigonometric Functions
69. 69. 2. Graphs of Trigonometric Functions
70. 70. 2. Graphs of Trigonometric Functions y = sin x
71. 71. 2. Graphs of Trigonometric Functions y = sin x y = cos x
72. 72. 2. Graphs of Trigonometric Functions y = sin x y = cos x 1 y = csc x = sin x
73. 73. 2. Graphs of Trigonometric Functions y = sin x y = cos x 1 y = csc x = sin x y = csc x
74. 74. 2. Graphs of Trigonometric Functions y = sin x y = cos x 1 y = csc x = sin x y = csc x