1. Trigonometry-7 Trigonometry in Four Quadrants

2. Trigonometry The Four Quadrants Co-ordinates in the First Quadrant Trig Ratios in the First Quadrant Co-ordinates as angle θ decreases Trig Ratios of 0° Co-ordinates as angle θ increases Trig Ratios of 90°

3. The Four Quadrants 00 90  180  270  360 

4. THE FOUR QUADRANTS. X axis Y axis Take a pizza and cut it into four equal pieces. Each piece is one quarter of the whole pizza. First Quarter Second Quarter Fourth Quarter Third Quarter Now take the pizza away.

5. THE FOUR QUADRANTS. O X axis Y axis First Quarter Second Quarter Fourth Quarter Third Quarter In trigonometry we work with right angled triangles and the four quarters. We call the quarters quadrants.

6. y r THE FOUR QUADRANTS. O X axis Y axis First Quadrant Second Quadrant Fourth Quadrant Third Quadrant In trigonometry we work with right angled triangles and the four quarters. We call the quarters quadrants.

7. THE FOUR QUADRANTS. O X axis One full revolution is 360 . 00 One quarter revolution is 90 . 90  180  270  360  4x90  =360 

8. THE FOUR QUADRANTS. O X axis All the work we will do in this lesson is in the first quadrant. 00 We will work from 0  to 90 . 90  180  270  360 

9. Well Done!

10. Co-ordinates in the First Quadrant y x A(x, y) θ

11. The Basic Trig Triangle  In triangle ABC, AB = 5 units, BC = 3 units and AC = 4 units.  How would you prove that triangle ABC is right angled? A B C 3 4 5 Did someone say Pythagoras?

12. The Basic Trig Triangle  Pythagoras’s theorem states:  The square on the hypotenuse equals the sum of the squares on the other two sides. A B C 3 4 5 Is AB 2 = AC 2 + BC 2 ? 5 2 = 4 2 + 3 2 25 = 16 + 9 TRUE This means ABC is right angled. In this triangle:

13. 3 4 5 Draw the basic 3, 4, 5 triangle Draw a circle around it The radius is the length of the hypotenuse

14. 3 4 5 The centre of the circle is at the letter O Point O is called the origin Draw axes through the origin O X axis Y axis What are the co-ordinates of this point?

15. 3 4 5 How far do we move along the X and Y axis to get to the point? Move 3 along Y axis Move 4 along X axis. O X axis Y axis What are the co-ordinates of this point? Co-ordinates of the point are: (4, 3)

16. 3 4 5 How far do we move along the X and Y axis to get to the point? Move 4 along Y axis Move 3 along X axis. O X axis Y axis What are the co-ordinates of this point? Co-ordinates of the point are: (3, 4)

17. y x The co-ordinates of point A are (4,3) What are the values of x and y in the diagram? O X axis Y axis A(, )43 x is the distance along the X axis to get to the point. x = 4

18. y x The co-ordinates of point A are (4,3) What are the values of x and y in the diagram? O X axis Y axis A(, )43 y is the distance along the Y axis to get to the point. y = 3

19. y x The co-ordinates of point A are (3,4) What are the values of x and y in the diagram? O X axis Y axis A(, )34 x is the distance along the X axis to get to the point. x = 3

20. y x The co-ordinates of point A are (3,4) What are the values of x and y in the diagram? O X axis Y axis A(, )34 y is the distance along the Y axis to get to the point. y = 4

21. y x r Now you know what co-ordinates are. O X axis Y axis What are the co-ordinates of this point? Let’s look at a general case.

22. y x r How far do we move along the X and Y axis to get to the point? Move y along Y axis Move x along X axis. O X axis Y axis Co-ordinates of the point are: (x, y) What are the co-ordinates of this point?

23. Well Done!

24. Trig Ratios in the First Quadrant Tan  Sin θ Cos θ Sine θ Cosine θ Tangent 

25. y x r In the diagram, can you work out the trig ratios for angle  in terms of the sides of the triangle? O X axis Y axis Co-ordinates of the point are: (x, y)  We need to label the opposite and adjacent sides, and the hypotenuse.

26. y x r In the diagram, can you work out the trig ratios for angle  in terms of the sides of the triangle? O X axis Y axis (x, y)  We need to label the opposite and adjacent sides, and the hypotenuse. O

27. y x r In the diagram, can you work out the trig ratios for angle  in terms of the sides of the triangle? O X axis Y axis (x, y)  We need to label the opposite and adjacent sides, and the hypotenuse. O H A O A Tan  = O H Sin  = A H Cos  = y x = y r = x r =

28. y x r Now you know how to write down the trig ratios of angle  in terms of the sides x, y and r of a right angled triangle. O X axis Y axis (x, y)  O A Tan  = y x = O H Sin  = y r = A H Cos  = x r =

29. Well Done!

30. Co-ordinates as Angle θ gets smaller A(x, y) 

31. What happens to x and y as angle  decreases? O X axis Start with point A(x,y) in the first quadrant. 00 What happens when angle  gets smaller? 90  y x r A(x, y)  Will the radius r change? Yes No

32. Oops! Click to try again. When you draw a circle the compass width stays the same.

33. Well Done!

34. What happens to x and y as angle  decreases? O X axis Watch what happens when  changes. 00 90  y x r A(x, y) 

35. What happens to x and y as angle  decreases? O X axis Watch what happens when  changes. 00 90  y x r A(x, y) 

36. What happens to x and y as angle  decreases? O X axis Watch what happens when  changes. 00 90  y x r A(x, y) 

37. What happens to x and y as angle  decreases? O X axis Watch what happens when  changes. 00 90  y x r A(x, y) 

38. What happens to x and y as angle  decreases? O X axis Watch what happens when  changes. 00 90  y x r A(x, y) 

39. What happens to x and y as angle  decreases? O X axis Watch what happens when  changes. 00 90  y x r A(x, y) 

40. What happens to x and y as angle  decreases? O X axis Watch what happens when  changes. 00 90  y x r A(x, y)  When  = 0 , how big is side y? y = r y = 0 y = x

41. Oops! Click to try again. Look at the slides again and see what happens to the length of y.

42. Well Done!

43. What happens to x and y as angle  decreases? O X axis Watch what happens when  changes. 00 90  y x r A(x, y)  When  = 0 , how big is side x? x = r x = 0 x = y

44. Oops! Click to try again. Look at the slides again and see what happens to the length of x.

45. Well Done!

46. The Trig Ratios of 0° A(x, y) 0°

47. The Trigonometric ratios of 0  O X axis What is the value of the following trig ratio? 00 90  y x r A(x, y)  0 r 1 Opp Adj Tan  =

48. Oops! Click to try again. = Opp Adj Tan  = y x = 0 r =0

49. Well Done!

50. The Trigonometric ratios of 0  O X axis What is the value of the following trig ratio? 00 90  y x r A(x, y)  0 r 1 Opp Hyp Sin  =

51. Oops! Click to try again. = Opp Hyp Sin  = y r = 0 r =0

52. Well Done!

53. The Trigonometric ratios of 0  O X axis What is the value of the following trig ratio? 00 90  y x r A(x, y)  0 r 1 Adj Hyp Cos  =

54. Oops! Click to try again. = Adj Hyp Cos  = x r = r r =1

55. Well Done!

56. Co-ordinates as Angle θ gets Bigger A(x, y) 

57. What happens to x and y as angle  increases? O X axis Start with point A(x,y) in the first quadrant. 00 What happens when angle  gets bigger? 90  y x r A(x, y)  Will the radius r change? Yes No

58. Oops! Click to try again. When you draw a circle the compass width stays the same.

59. Well Done!

60. What happens to x and y as angle  increases? O X axis Watch what happens when  changes. 00 90  y x r A(x, y) 

61. What happens to x and y as angle  increases? O X axis Watch what happens when  changes. 00 90  y x r A(x, y) 

62. What happens to x and y as angle  increases? O X axis Watch what happens when  changes. 00 90  y x r A(x, y) 

63. What happens to x and y as angle  increases? O X axis Watch what happens when  changes. 00 90  y x r A(x, y) 

64. What happens to x and y as angle  increases? O X axis Watch what happens when  changes. 00 90  y x r A(x, y) 

65. What happens to x and y as angle  increases? O X axis Watch what happens when  changes. 00 90  y x r A(x, y)  When  = 90 , how big is side y? y = r y = 0 y = x

66. Oops! Click to try again. Look at the slides again and see what happens to the length of y.

67. Well Done!

68. What happens to x and y as angle  increases? O X axis Watch what happens when  changes. 00 90  y x r A(x, y)  When  = 90 , how big is side x? x = r x = 0 x = y

69. Oops! Click to try again. Look at the slides again and see what happens to the length of x.

70. Well Done!

71. X axis 00 90  y x r  The Trig Ratios of 90°

72. O X axis What is the value of the following trig ratio? 00 90  y x r A(x, y)  0 1 undefined The Trig Ratios of 90° Opp Adj Tan  =

73. Oops! Click to try again. = Opp Adj Tan  = y x = ∞ You can’t divide by 0. = r 0

74. Well Done!

75. O X axis What is the value of the following trig ratio? 00 90  y x r A(x, y)  0 1 undefined The Trig Ratios of 90° Opp Hyp Sin  =

76. Oops! Click to try again. = Opp Hyp Sin  = y r = r r =1

77. Well Done!

78. O X axis What is the value of the following trig ratio? 00 90  y x r A(x, y)  0 1 undefined The Trig Ratios of 90° Adj Hyp Cos  =

79. Oops! Click to try again. = Adj Hyp Cos  = x r = 0 r =0

80. Well Done!