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Graphs of the form y = a sin x o

Trigonometry Graphs . National 5. Exact values for Sin Cos and Tan. Angles greater than 90 o. Graphs of the form y = a sin x o. Graphs of the form y = a sin bx o. www.mathsrevision.com. Graphs of the form y = a sin bx o + c. Solving Trig Equations. Special trig relationships. National 5.

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Graphs of the form y = a sin x o

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  1. Trigonometry Graphs National 5 Exact values for Sin Cos and Tan Angles greater than 90o Graphs of the form y = a sin xo Graphs of the form y = a sin bxo www.mathsrevision.com Graphs of the form y = a sin bxo + c Solving Trig Equations Special trig relationships created by Mr. Lafferty

  2. National 5 Starter Questions www.mathsrevision.com Created by Mr Lafferty Maths Dept

  3. National 5 Exact Values Learning Intention Success Criteria • Recognise basic triangles and exact values for sin, cos and tan 30o, 45o, 60o . • To build on basic trigonometry values. • Calculate exact values for problems. www.mathsrevision.com Created by Mr Lafferty Maths Dept

  4. 60º 2 2 2 60º 60º 60º 2 Exact Values National 5 Some special values of Sin, Cos and Tan are useful left as fractions, We call these exact values 30º 3 www.mathsrevision.com 1 This triangle will provide exact values for sin, cos and tan 30º and 60º

  5. Exact Values National 5 3 2 ½ 1 0 www.mathsrevision.com 3 2 1 ½ 0 0 3

  6. Exact Values National 5 Consider the square with sides 1 unit 45º 2 1 1 www.mathsrevision.com 45º 1 1 We are now in a position to calculate exact values for sin, cos and tan of 45o

  7. Exact Values National 5 3 2 1 2 ½ 1 0 www.mathsrevision.com 3 2 1 2 1 ½ 0 0 1 3

  8. Exact Values National 5 Now try Ex 2.1 Ch11 (page 220) www.mathsrevision.com Created by Mr Lafferty Maths Dept

  9. National 5 Starter Questions www.mathsrevision.com Created by Mr Lafferty Maths Dept

  10. Angles Greater than 90o National 5 Learning Intention Success Criteria • Find values of sine, cosine and tangent over the range 0o to 360o. • Introduce definition of sine, cosine and tangent over 360o using triangles with the unity circle. • 2. Recognise the symmetry and equal values for sine, cosine and tangent. www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

  11. r y x P(x,y) y x Angles Greater than 90o National 5 We will now use a new definition to cater for ALL angles. Demo Sin Demo Cos Demo Tan New Definitions y-axis www.mathsrevision.com r Ao x-axis O www.mathsrevision.com

  12. Trigonometry Angles over 900 Example National 5 The radius line is 2cm. The point (1.2, 1.6). Find sin cos and tan for the angle. (1.2, 1.6) Check answer with calculator www.mathsrevision.com 53o Created by Mr Lafferty Maths Dept

  13. Trigonometry Angles over 900 Example 1 National 5 Check answer with calculator The radius line is 2cm. The point (-1.8, 0.8). Find sin cos and tan for the angle. (-1.8, 0.8) www.mathsrevision.com 127o Created by Mr Lafferty Maths Dept

  14. Summary of results Trigonometry All Quadrants Example National 5 Calculate the ration for sin cos and tan for the angle values below. 90o 30o 210o 45o 225o Sin +ve All +ve 60o 240o xo 180o - xo 120o 300o 0o www.mathsrevision.com 180o 135o 315o 180o + xo 360o - xo 150o 330o Cos +ve Tan +ve 270o Created by Mr Lafferty Maths Dept

  15. What Goes In The Box ? National 5 Write down the equivalent values of the following in term of the first quadrant (between 0o and 90o): • Sin 300o • Cos 360o • Tan 330o • Sin 380o • Cos 460o • Sin 135o • Cos 150o • Tan 135o • Sin 225o • Cos 270o - sin 60o sin 45o cos 0o -cos 45o www.mathsrevision.com - tan 30o -tan 45o sin 20o -sin 45o - cos 80o -cos 90o

  16. Trigonometry Angles over 900 National 5 Now try MIA Ch11 Ex3.1 Ch11 (page 222) www.mathsrevision.com Created by Mr Lafferty Maths Dept

  17. National 5 Starter www.mathsrevision.com created by Mr. Lafferty

  18. Sine Graph National 5 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

  19. Key Features Sine Graph Zeros at 0, 180o and 360o Max value at x = 90o National 5 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

  20. 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 National 5 3 2 1 0 www.mathsrevision.com 90o 180o 270o 360o -1 What effect does the negative sign have on the graphs ? -2 -3 Demo created by Mr. Lafferty

  21. Sine Graph National 5 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

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

  23. Key Features Cosine Graphs Zeros at 90o and 270o Max value at x = 0o and 360o National 5 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

  24. What effect does the number at the front have on the graphs ? y = cosxo y = 2cosxo y = 3cosxo y = 0.5cosxo y = -cosxo Cosine National 5 3 2 1 0 www.mathsrevision.com 90o 180o 270o 360o -1 -2 Demo -3 created by Mr. Lafferty

  25. y = 2cosxo y = 4cosxo y = 6cosxo y = 0.5cosxo y = -cosxo Cosine Graph National 5 6 4 2 0 www.mathsrevision.com 90o 180o 270o 360o -2 -4 -6 created by Mr. Lafferty

  26. Key Features Tangent Graphs Zeros at 0 and 180o National 5 Key Features www.mathsrevision.com Domain is 0 to 180o (repeats itself every 180o) created by Mr. Lafferty

  27. Tangent Graphs National 5 www.mathsrevision.com Demo created by Mr. Lafferty

  28. Tangent Graph National 5 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

  29. Period of a Function National 5 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

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

  31. Trigonometry Graphs National 5 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

  32. y = cosxo y = cos2xo y = cos3xo Cosine National 5 3 2 1 0 www.mathsrevision.com 90o 180o 270o 360o -1 -2 Demo -3 created by Mr. Lafferty

  33. Trigonometry Graphs National 5 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

  34. Trigonometry Graphs National 5 y = a tan (bx) How many times it repeats itself in 180o Demo 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

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

  36. Write down equations for the graphs shown? y = 1.5cos2xo y = -2cos2xo y = 0.5cos4xo Cosine Combinations National 5 3 2 1 0 www.mathsrevision.com 90o 180o 270o 360o -1 -2 -3 created by Mr. Lafferty

  37. Combination Graphs National 5 Now Try MIA Ch11 Ex 5.1 Page 227 www.mathsrevision.com created by Mr. Lafferty

  38. C moves the graph up or down in the y-axis direction Trigonometry Graphs National 5 y = a sin (bx) + c Demo How many times it repeats itself in 360o a - Amplitude 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

  39. Sine Graph Simply move graph up by 1 National 5 1 0 www.mathsrevision.com 45o 90o 180o 270o 360o Given the basic y = sin x graph what does the graph of y = sin x + 1 look like? -1 created by Mr. Lafferty

  40. Given the y = cos x graph. What does the graph of y = cos x – 0.5 look like? Cosine Graph Simply move down by 0.5 National 5 1 0 160o www.mathsrevision.com 90o 180o 270o 360o -1 created by Mr. Lafferty

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

  42. Write down equations for the graphs shown? Cosine y = cos2xo + 1 y = -2cos2xo - 1 Combinations National 5 3 2 1 0 www.mathsrevision.com 90o 180o 270o 360o -1 -2 -3 created by Mr. Lafferty

  43. Combination Graphs National 5 Now try MIA Ch11 Ex 5.2 Page 231 www.mathsrevision.com created by Mr. Lafferty

  44. National 5 Starter www.mathsrevision.com created by Mr. Lafferty

  45. Solving Trig Equations National 5 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

  46. Solving Trig Equations National 5 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

  47. Solving Trig Equations Graphically what are we trying to solve a sin xo + b = 0 National 5 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

  48. Solving Trig Equations Graphically what are we trying to solve a sin xo + b = 0 National 5 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

  49. Solving Trig Equations Graphically what are we trying to solve a cos xo + b = 0 National 5 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

  50. Solving Trig Equations Graphically what are we trying to solve a tan xo + b = 0 National 5 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

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