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Graphs of the form y = a sin x o

Trigonometry Graphs. Int 2. Graphs of the form y = a sin x o. Graphs of the form y = a sin bx o. Phase angle. www.mathsrevision.com. Solving Trig Equations. Special trig relationships. Int 2. Starter. www.mathsrevision.com. Sine Graph. Int 2. Learning Intention. Success Criteria.

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Graphs of the form y = a sin x o

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  1. Trigonometry Graphs Int 2 Graphs of the form y = a sin xo Graphs of the form y = a sin bxo Phase angle www.mathsrevision.com Solving Trig Equations Special trig relationships created by Mr. Lafferty

  2. Int 2 Starter www.mathsrevision.com created by Mr. Lafferty

  3. Sine Graph Int 2 Learning Intention Success Criteria • Identify the key points for various graphs. • To investigate graphs of the form • y = a sin xo • y = a cos xo • y = tan xo www.mathsrevision.com created by Mr. Lafferty

  4. Key Features Sine Graph Zeros at 0, 180o and 360o Max value at x = 90o Int 2 Minimum value at x = 270o Key Features www.mathsrevision.com Domain is 0 to 360o (repeats itself every 360o) Maximum value of 1 Minimum value of -1 created by Mr. Lafferty

  5. What effect does the number at the front have on the graphs ? y = sinxo y = 2sinxo y = 3sinxo y = 0.5sinxo y = -sinxo Sine Graph Int 2 3 2 1 0 www.mathsrevision.com 90o 180o 270o 360o -1 -2 -3 created by Mr. Lafferty

  6. Sine Graph Int 2 y = a sin (x) www.mathsrevision.com For a > 1 stretches graph in the y-axis direction For a < 1 compresses graph in the y - axis direction For a - negative flips graph in the x – axis. created by Mr. Lafferty

  7. y = 5sinxo y = 4sinxo y = sinxo y = -6sinxo Sine Graph Int 2 6 4 2 0 www.mathsrevision.com 90o 180o 270o 360o -2 -4 -6 created by Mr. Lafferty

  8. Key Features Cosine Graphs Zeros at 90o and 270o Max value at x = 0o and 360o Int 2 Minimum value at x = 180o Key Features www.mathsrevision.com Domain is 0 to 360o (repeats itself every 360o) Maximum value of 1 Minimum value of -1 created by Mr. Lafferty

  9. What effect does the number at the front have on the graphs ? y = cosxo y = 2cosxo y = 3cosxo y = cosxo y = -cosxo Cosine Int 2 3 2 1 0 www.mathsrevision.com 90o 180o 270o 360o -1 -2 -3 created by Mr. Lafferty

  10. y = cosxo y = 4cosxo y = 6cosxo y = cosxo y = -cosxo Cosine Graph Int 2 6 4 2 0 www.mathsrevision.com 90o 180o 270o 360o -2 -4 -6 created by Mr. Lafferty

  11. Key Features Tangent Graphs Zeros at 0 and 180o Int 2 Key Features www.mathsrevision.com Domain is 0 to 180o (repeats itself every 180o) created by Mr. Lafferty

  12. Tangent Graphs Int 2 www.mathsrevision.com created by Mr. Lafferty

  13. Tangent Graph Int 2 y = a tan (x) www.mathsrevision.com For a > 1 stretches graph in the y-axis direction For a < 1 compresses graph in the y - axis direction For a - negative flips graph in the x – axis. created by Mr. Lafferty

  14. Combination Graphs Int 2 Revision Booklet All questions www.mathsrevision.com created by Mr. Lafferty

  15. Int 2 Starter www.mathsrevision.com created by Mr. Lafferty

  16. Trig Graphs Int 2 Learning Intention Success Criteria • Identify the key points for more complicated Trig graphs. • To investigate graphs of the form • y = a sin bxo • y = a cos bxo • y = tan bxo www.mathsrevision.com created by Mr. Lafferty

  17. Period of a Function Int 2 When a pattern repeats itself over and over, it is said to be periodic. Sine function has a period of 360o www.mathsrevision.com Let’s investigate the function y = sin bx created by Mr. Lafferty

  18. What effect does the number in front of x have on the graphs ? y = sinxo y = sin2xo y = sin4xo y = sin0.5xo Sine Graph Int 2 3 2 1 0 www.mathsrevision.com 90o 180o 270o 360o -1 -2 -3 created by Mr. Lafferty

  19. Trigonometry Graphs Int 2 y = a sin (bx) How many times it repeats itself in 360o www.mathsrevision.com For a > 1 stretches graph in the y-axis direction For a < 1 compresses graph in the y - axis direction For a - negative flips graph in the x – axis. created by Mr. Lafferty

  20. What effect does the number at the front have on the graphs ? y = cosxo y = cos2xo y = cos3xo Cosine Int 2 3 2 1 0 www.mathsrevision.com 90o 180o 270o 360o -1 -2 -3 created by Mr. Lafferty

  21. Trigonometry Graphs Int 2 y = a cos (bx) How many times it repeats itself in 360o www.mathsrevision.com For a > 1 stretches graph in the y-axis direction For a < 1 compresses graph in the y - axis direction For a - negative flips graph in the x – axis. created by Mr. Lafferty

  22. Trigonometry Graphs Int 2 y = a tan (bx) How many times it repeats itself in 180o www.mathsrevision.com For a > 1 stretches graph in the y-axis direction For a < 1 compresses graph in the y - axis direction For a - negative flips graph in the x – axis. created by Mr. Lafferty

  23. Write down equations for graphs shown ? y = 0.5sin2xo y = 2sin4xo y = 3sin0.5xo Trig Graph Combinations Int 2 3 2 1 0 www.mathsrevision.com 90o 180o 270o 360o -1 -2 -3 created by Mr. Lafferty

  24. Write down equations for the graphs shown? y = 1.5cos2xo y = -2cos2xo y = 0.5cos4xo Cosine Combinations Int 2 3 2 1 0 www.mathsrevision.com 90o 180o 270o 360o -1 -2 -3 created by Mr. Lafferty

  25. Combination Graphs Int 2 Revision Booklet All questions www.mathsrevision.com created by Mr. Lafferty

  26. Int 2 Starter www.mathsrevision.com created by Mr. Lafferty

  27. Phase Angle Int 2 Learning Intention Success Criteria • Understand the term phase angle / phase shift. • Read off the values • for a and b for a graph of the form. • y = a sin( x – c )o • To explain what phase angle / phase shift is using knowledge from quadratics. www.mathsrevision.com created by Mr. Lafferty

  28. By how much do we have to move the standard sine curve so it fits on the other sine curve? Sine Graph Int 2 y = sin(x - 45)o 1 To the right “-” 45o 0 www.mathsrevision.com 45o 90o 180o 270o 360o -1 created by Mr. Lafferty

  29. By how much do we have to move the standard sine curve so it fits on the other sine curve? Sine Graph Int 2 y = sin(x + 60)o 1 To the left “+” 60o 0 www.mathsrevision.com -60o 90o 180o 270o 360o -1 created by Mr. Lafferty

  30. Phase Angle Int 2 y = sin (x - c) Moves graph along x - axis www.mathsrevision.com For c > 0 moves graph to the right along x – axis For c < 0 moves graph to the left along x – axis created by Mr. Lafferty

  31. By how much do we have to move the standard cosine curve so it fits on the other cosine curve? Cosine Graph Int 2 y = cos(x - 70)o 1 To the right “-” 70o 0 160o www.mathsrevision.com 90o 180o 270o 360o -1 created by Mr. Lafferty

  32. By how much do we have to move the standard cosine curve so it fits on the other cosine curve? Cosine Graph Int 2 y = cos(x + 56)o 1 To the left “+” 56o 0 34o www.mathsrevision.com 90o 180o 270o 360o -1 created by Mr. Lafferty

  33. Summary of work So far Int 2 y = a sin (x - b) For a > 1 stretches graph in the y-axis direction For b > 0 moves graph to the right along x – axis For a < 1 compresses graph in the y - axis direction For b < 0 moves graph to the left along x – axis www.mathsrevision.com For a - negative flips graph in the x – axis. created by Mr. Lafferty

  34. Sketch Graph y = a cos (x – b) Int 2 a =3 b =30 y = 2 cos (x - 30) www.mathsrevision.com created by Mr. Lafferty

  35. Combination Graphs Int 2 Revision Booklet All questions www.mathsrevision.com created by Mr. Lafferty

  36. Int 2 Starter www.mathsrevision.com created by Mr. Lafferty

  37. Solving Trig Equations Int 2 Learning Intention Success Criteria • Use the rule for solving any ‘ normal ‘ equation • Realise that there are many solutions to trig equations depending on domain. • To explain how to solve • trig equations of the form • a sin xo + 1 = 0 www.mathsrevision.com created by Mr. Lafferty

  38. Solving Trig Equations Int 2 Sin +ve All +ve 180o - xo 180o + xo 360o - xo www.mathsrevision.com Cos +ve Tan +ve 1 2 3 4 created by Mr. Lafferty

  39. Solving Trig Equations Graphically what are we trying to solve a sin xo + b = 0 Int 2 Example 1 : Solving the equation sin xo = 0.5 in the range 0o to 360o sin xo = (0.5) xo = sin-1(0.5) www.mathsrevision.com xo = 30o There is another solution xo = 150o 1 2 3 4 (180o – 30o = 150o) created by Mr. Lafferty

  40. Solving Trig Equations Graphically what are we trying to solve a sin xo + b = 0 Int 2 Example 1 : Solving the equation 3sin xo + 1= 0 in the range 0o to 360o sin xo = -1/3 Calculate first Quad value xo = 19.5o www.mathsrevision.com x = 180o + 19.5o = 199.5o There is another solution 1 2 3 4 ( 360o - 19.5o = 340.5o) created by Mr. Lafferty

  41. Solving Trig Equations Graphically what are we trying to solve a cos xo + b = 0 Int 2 Example 1 : Solving the equation cos xo = 0.625 in the range 0o to 360o cos xo = 0.625 xo = cos -1 0.625 www.mathsrevision.com xo = 51.3o There is another solution (360o - 53.1o = 308.7o) 1 2 3 4 created by Mr. Lafferty

  42. Solving Trig Equations Graphically what are we trying to solve a tan xo + b = 0 Int 2 Example 1 : Solving the equation tan xo = 2 in the range 0o to 360o tan xo = 2 xo = tan -1(2) www.mathsrevision.com xo = 63.4o There is another solution x = 180o + 63.4o = 243.4o 1 2 3 4 created by Mr. Lafferty

  43. Solving Trig Equations Int 2 Now try MIA Ex6 First Column Only (page 249) www.mathsrevision.com created by Mr. Lafferty

  44. Int 2 Starter www.mathsrevision.com created by Mr. Lafferty

  45. Solving Trig Equations Int 2 Learning Intention Success Criteria • Know and learn the two special trig relationships. • Apply them to solve problems. • To explain some special trig relationships • sin 2 xo +cos 2xo = ? • and • tan xo and sin x • cos x www.mathsrevision.com created by Mr. Lafferty

  46. Solving Trig Equations Int 2 Lets investigate sin 2xo + cos 2 xo = ? Calculate value for x = 10, 20, 50, 250 www.mathsrevision.com Learn ! sin 2xo + cos 2 xo = 1 created by Mr. Lafferty

  47. sin xo sin xo cos xo cos xo Solving Trig Equations Int 2 Lets investigate tan xo and Calculate value for x = 10, 20, 50, 250 www.mathsrevision.com Learn ! tan xo = created by Mr. Lafferty

  48. Solving Trig Equations Int 2 Now try MIA Ex7 (page 252) www.mathsrevision.com created by Mr. Lafferty

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