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E-learning

This e-learning module focuses on chapter 11, where we explore the graphical representations of the sine, cosine, and tangent functions. We discuss their periodic nature, amplitude, and significant points such as max/min values, axis intersections, and asymptotes for the tangent function. Additionally, we provide examples on how to sketch transformed trigonometric functions, including reflections and translations. This comprehensive guide will enhance your understanding of trigonometric graphs, which are fundamental in mathematics and various applications.

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E-learning

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  1. E-learning extended learning for chapter 11 (graphs)

  2. Let’s recall first

  3. Graph of y = sin  Note: max value = 1 and min value = -1 Graph of y = sin 

  4. Graph of y = sin  The graph will repeats itself for every 360˚. The length of interval which the curve repeats is call the period. Therefore, sine curve has a period of 360˚. Graph of y = sin 

  5. Graph of y = sin  2 Graph of y = sin  2 - 2

  6. In general, graph of y = sin  a Graph of y = sin  a - a

  7. Graph of y = cos  Note: max value = 1 and min value = -1

  8. Graph of y = cos  3 3 - 3

  9. In general, graph of y = cos  a a - a

  10. Graph of y = cos  The graph will repeats itself for every 360˚. Therefore, cosine curve has a period of 360˚.

  11. Graph of y = tan  45˚ 225˚ 135˚ 315˚ Note: The graph is not continuous. There are break at 90˚ and 270˚. The curve approach the line at 90˚ and 270˚. Such lines are called asymptotes.

  12. In general, graph of y = tan  a Note: The graph does not have max and min value. a - a

  13. Summary Identify the 3 types of graphs: y = sin  y = cos  y = tan 

  14. Points to consider when sketchingtrigonometrical functions: • Easily determined points: a) maximum and minimum points b) points where the graph cuts the axes • Period of the function • Asymptotes (for tangent function) 14

  15. Let’s continue learning..

  16. Example 1: Sketch y = 4sin x (given y = sin x )for 0° x  360° x y = 4sin x x x x x x x x > x x y = sin x x Comparing the 2 graphs, what happens to the max and min point of y = 4 sin x?

  17. Example 2: Sketch y = 4 + sin x for 0° x  360° x y = 4 + sin x x x x x x x x x > x x y = sin x Spot the difference between y = 4 sin x and y = 4 + sin x and write down the answer.

  18. Example 3: Sketch y = - sin x for 0° x  360° How do we get y = - sin x graph from y = sin x? x x y = - sin x x > x x x x x x y = sin x x x Reflection of y = sin x in x axis

  19. Example 4: Sketch y = 4 - sin x for 0° x  360° x x x x y = 4 + (- sin x) x y = - sin x > x y = sin x • Reflection of y = sin x in x axis • Translation of y = -sin x by 4 units along y axis

  20. Example 5: Sketch y = |sin x| for 0° x  360° y = |sin x| > x y = sin x

  21. Example 6: Sketch y = -|sin x| for 0° x  360° y = |sin x| > x y = -|sin x| • Reflection of y = |sin x| in x axis

  22. Example 7 Sketch y = -5cos x for 0° x  360° Reflection about x axis x x 5 -5 x x x x y = cos x y = 5cos x y = -5cos x

  23. Example 8 Sketch y = 3 + tan x for 0° x  360° x x x x x x x x y = tan x y = 3 + tan x

  24. Example 9 Sketch y = 2 – sin x, for values of x between 0° x  360° y = sin x y = - sin x y = 2 – sin x

  25. Example 10 Sketch y = 1 – 3cos x for values of x between 0° x  360° y = 3 cos x y = cos x y = 3 cos x y = - 3 cos x y = 1 – 3 cos x

  26. Example 11 Sketch y = |3cos x| for values of x between 0° x  360° y = 3 cos x y = |3 cos x|

  27. Example 12 Sketch y = |3 sin x| - 2 for values of x between 0° x  360° y = 3 sin x y = sin x y = 3 sin x y = |3 sin x| y = y = |3 sin x| - 2

  28. Example 13 Sketch y = 2cos x -1 and y = -2|sin x| for values of x between 0  x  360. Hence find the no. of solutions 2cos x -1 = -2|sin x| in the interval. Solution: Answer: No of solutions = 2 y = 2cos x -1 x x y = -2|sin x|

  29. Example 14 Sketch y = |tan x| and y = 1 - sin x for values of x between 0  x  360. Hence find the no. of solutions |tan x| = 1 - sin x| in the interval. Solution: Answer: No of solutions = 4 y = |tan x| x x y = 1 - sin x x x

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