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

2. 9-Aug-15Created by Mr Lafferty Maths Dept Starter Questions www.mathsrevision.com National 5

3. 9-Aug-15Created by Mr Lafferty Maths Dept Exact Values Learning Intention Success Criteria 1.Recognise basic triangles and exact values for sin, cos and tan 30 o, 45 o, 60 o. 1.To build on basic trigonometry values. 2.Calculate exact values for problems. www.mathsrevision.com National 5

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

5. x0º30º45º60º90º Sin xº Cos xº Tan xº  ½ ½ 33  3 2  3 2 0 1 0 1 0 Exact Values www.mathsrevision.com National 5

6. Exact Values www.mathsrevision.com 11 45º 2 2 Consider the square with sides 1 unit 1 1 We are now in a position to calculate exact values for sin, cos and tan of 45 o National 5

7. x0º30º45º60º90º Sin xº Cos xº Tan xº  ½ ½ 33  3 2  3 2 0 1 0 1 0 Exact Values www.mathsrevision.com National 5 1  2 1

8. 9-Aug-15Created by Mr Lafferty Maths Dept Now try Ex 2.1 Ch11 (page 220) www.mathsrevision.com Exact Values National 5

9. 9-Aug-15Created by Mr Lafferty Maths Dept Starter Questions www.mathsrevision.com National 5

10. 9-Aug-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Learning Intention Success Criteria Angles Greater than 90 o 1.Introduce definition of sine, cosine and tangent over 360 o using triangles with the unity circle. 1.Find values of sine, cosine and tangent over the range 0 o to 360 o. 2.Recognise the symmetry and equal values for sine, cosine and tangent. National 5

11. 9-Aug-15www.mathsrevision.com 11 x y r Angles Greater than 90 o www.mathsrevision.com We will now use a new definition to cater for ALL angles. O x-axis r y-axis y x A o New Definitions P(x,y) National 5 Demo SinDemo CosDemo Tan

12. 9-Aug-15Created by Mr Lafferty Maths Dept www.mathsrevision.com Trigonometry Angles over 90 0 (1.2, 1.6) 53 o The radius line is 2cm. The point (1.2, 1.6). Find sin cos and tan for the angle. Check answer with calculator Example National 5

13. 9-Aug-15Created by Mr Lafferty Maths Dept www.mathsrevision.com Trigonometry Angles over 90 0 The radius line is 2cm. The point (-1.8, 0.8). Find sin cos and tan for the angle. Check answer with calculator Example 1 National 5 (-1.8, 0.8) 127 o

14. 9-Aug-15Created by Mr Lafferty Maths Dept www.mathsrevision.com Trigonometry All Quadrants Calculate the ration for sin cos and tan for the angle values below. Summary of results Example National 5 All +veSin +ve Tan +ve Cos +ve 180 o - x o 180 o + x o 360 o - x o 30 o 60 o 150 o 120 o 45 o 135 o 210 o 240 o 330 o 300 o 225 o 315 o Sin xCos xTan x 90 o 180 o 270 o 0 o x o

15. 1)Sin 135 o 2)Cos 150 o 3)Tan 135 o 4)Sin 225 o 5)Cos 270 o What Goes In The Box ? Write down the equivalent values of the following in term of the first quadrant (between 0 o and 90 o ): sin 45 o www.mathsrevision.com 1)Sin 300 o 2)Cos 360 o 3)Tan 330 o 4)Sin 380 o 5)Cos 460 o -cos 45 o -tan 45 o -sin 45 o -cos 90 o - sin 60 o cos 0 o - tan 30 o sin 20 o - cos 80 o National 5

16. 9-Aug-15Created by Mr Lafferty Maths Dept Now try MIA Ch11 Ex3.1 Ch11 (page 222) www.mathsrevision.com Trigonometry Angles over 90 0 National 5

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

18. created by Mr. Lafferty Learning Intention Success Criteria 1.Identify the key points for various graphs. 1.To investigate graphs of the form y = a sin x o y = a cos x o y = tan x o www.mathsrevision.com Sine Graph National 5

19. created by Mr. Lafferty www.mathsrevision.com Sine Graph Key Features Domain is 0 to 360 o (repeats itself every 360 o ) Maximum value of 1 Minimum value of -1 Key Features Zeros at 0, 180 o and 360 o Max value at x = 90 o Minimum value at x = 270 o National 5

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

21. created by Mr. Lafferty Sine Graph www.mathsrevision.com y = a sin (x) 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. National 5

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

23. created by Mr. Lafferty www.mathsrevision.com Cosine Graphs Key Features Key Features Domain is 0 to 360 o (repeats itself every 360 o ) Maximum value of 1 Minimum value of -1 Zeros at 90 o and 270 o Max value at x = 0 o and 360 o Minimum value at x = 180 o National 5

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

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

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

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

28. created by Mr. Lafferty Tangent Graph www.mathsrevision.com y = a tan (x) 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. National 5

29. created by Mr. Lafferty www.mathsrevision.com When a pattern repeats itself over and over, it is said to be periodic. Sine function has a period of 360 o Period of a Function Let’s investigate the function y = sin bx National 5

30. created by Mr. Lafferty www.mathsrevision.com Sine Graph 1 2 3 -3 -2 0 90 o 180 o 270 o 360 o y = sinx o y = sin2x o y = sin4x o y = sin0.5x o National 5 What effect does the number in front of x have on the graphs ? Demo

31. created by Mr. Lafferty www.mathsrevision.com Trigonometry Graphs y = a sin (bx) 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. How many times it repeats itself in 360 o National 5

32. created by Mr. Lafferty www.mathsrevision.com Cosine 1 2 3 -3 -2 0 90 o 180 o 270 o 360 o y = cosx o y = cos2x o y = cos3x o National 5 Demo

33. created by Mr. Lafferty www.mathsrevision.com Trigonometry Graphs y = a cos (bx) 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. How many times it repeats itself in 360 o National 5

34. created by Mr. Lafferty www.mathsrevision.com Trigonometry Graphs y = a tan (bx) 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. How many times it repeats itself in 180 o National 5 Demo

35. y = 0.5sin2x o y = 2sin4x o y = -3sin0.5x o created by Mr. Lafferty www.mathsrevision.com Trig Graph 1 2 3 -3 -2 0 90 o 180 o 270 o 360 o Write down the equations for the graphs shown ? Combinations National 5

36. created by Mr. Lafferty www.mathsrevision.com Cosine 1 2 3 -3 -2 0 90 o 180 o 270 o 360 o Combinations y = 1.5cos2x o y = -2cos2x o y = 0.5cos4x o National 5 Write down equations for the graphs shown?

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

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

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

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

41. created by Mr. Lafferty www.mathsrevision.com Trig Graph 1 2 3 -3 -2 0 90 o 180 o 270 o 360 o Write down equations for graphs shown ? Combinations National 5 y = 0.5sin2x o + 0.5 y = 2sin4x o - 1

42. created by Mr. Lafferty www.mathsrevision.com Cosine 1 2 3 -3 -2 0 90 o 180 o 270 o 360 o Combinations y = cos2x o + 1 y = -2cos2x o - 1 National 5 Write down equations for the graphs shown?

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

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

45. created by Mr. Lafferty Learning Intention Success Criteria 1.Use the rule for solving any ‘ normal ‘ equation 2.Realise that there are many solutions to trig equations depending on domain. 1.To explain how to solve trig equations of the form a sin x o + 1 = 0 www.mathsrevision.com Solving Trig Equations National 5

46. created by Mr. Lafferty www.mathsrevision.com Solving Trig Equations All +veSin +ve Tan +ve Cos +ve 180 o - x o 180 o + x o 360 o - x o 1234 National 5

47. created by Mr. Lafferty www.mathsrevision.com Solving Trig Equations a sin x o + b = 0 Example 1 : Solving the equation sin x o = 0.5 in the range 0 o to 360 o Graphically what are we trying to solve x o = sin -1 (0.5) x o = 30 o There is another solution x o = 150 o (180 o – 30 o = 150 o ) sin x o = (0.5) 1234 National 5

48. created by Mr. Lafferty www.mathsrevision.com Solving Trig Equations a sin x o + b = 0 Example 1 : Solving the equation 3sin x o + 1= 0 in the range 0 o to 360 o Graphically what are we trying to solve sin x o = -1/3 Calculate first Quad value x o = 19.5 o x = 180 o + 19.5 o = 199.5 o ( 360 o - 19.5 o = 340.5 o ) There is another solution 1234 National 5

49. created by Mr. Lafferty www.mathsrevision.com Solving Trig Equations a cos x o + b = 0 Example 1 : Solving the equation cos x o = 0.625 in the range 0 o to 360 o Graphically what are we trying to solve cos x o = 0.625 x o = 51.3 o (360 o - 53.1 o = 308.7 o ) x o = cos -1 0.625 There is another solution 1234 National 5

50. created by Mr. Lafferty www.mathsrevision.com Solving Trig Equations a tan x o + b = 0 Example 1 : Solving the equation tan x o = 2 in the range 0 o to 360 o Graphically what are we trying to solve tan x o = 2 x o = 63.4 o x = 180 o + 63.4 o = 243.4 o x o = tan -1 (2) There is another solution 1234 National 5

51. created by Mr. Lafferty Now try MIA Ch11 Ex6.1, 6.2 and 7.1 (page 236) www.mathsrevision.com Solving Trig Equations National 5

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

53. created by Mr. Lafferty Learning Intention Success Criteria 1.Know and learn the two special trig relationships. 2.Apply them to solve problems. 1.To explain some special trig relationships sin 2 x o + cos 2 x o = ? and tan x o and sin x cos x www.mathsrevision.com Solving Trig Equations National 5

54. created by Mr. Lafferty www.mathsrevision.com Solving Trig Equations Lets investigate sin 2 x o + cos 2 x o = ? Calculate value for x = 10, 20, 50, 250 sin 2 x o + cos 2 x o = 1 Learn ! National 5

55. created by Mr. Lafferty www.mathsrevision.com Solving Trig Equations Lets investigate tan x o Calculate value for x = 10, 20, 50, 250 Learn ! sin x o cos x o and tan x o sin x o cos x o = National 5

56. created by Mr. Lafferty Now try MIA Ex8.1 Ch11 (page 238) www.mathsrevision.com Solving Trig Equations National 5