2. www.mathsrevision.com Trigonometry National 4 REL 1.3a REL 1.3b Investigating the Tangent Ratio The Sine Ratio (Finding a length) The Sine Ratio (Finding an angle) The Cosine of an Angle Mixed Problems The Tangent (Finding lengths) The Tangent (Finding the Angle) The Cosine ( Finding a length)

3. Starter Questions 18-May-15Created by Mr.Lafferty Maths Dept www.mathsrevision.com Q1.Calculate Q2.Convert 60% to fraction and simplify. Q3.Convert Q4.What is the time difference 09:28 and 10:50 to a percentage. Q5.The answer to the question is 90. What is the question. National 4 REL 1.3a REL 1.3b

4. 18-May-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Learning Intention Success Criteria 2.Work out Tan Ratio. 1.To identify the hypotenuse, opposite and adjacent sides in a right angled triangle. Angles & Triangles 1.Understand the terms hypotenuse, opposite and adjacent in right angled triangle. National 4 REL 1.3a REL 1.3b

5. www.mathsrevision.com Trigonometry National 4 REL 1.3a REL 1.3b Trigonometry means “triangle” and “measurement”. Adjacent Opposite x°x°x°x° hypotenuse We will be using right-angled triangles.

6. These triangles have 2 things in common One is 37 o The other we will investigate

7. TriangleABCD Vertical side3cm6cm4cm Horizontal side4cm8cm6cm 1.5cm 2cm 0.75 0.75

8. www.mathsrevision.com Trigonometry For the RAT with angle of 37 o the answer obtained by dividing vertical side by horizontal side was Right-Angled Triangle ALWAYS 0.75 The number has a special name It is called the TANGENT of 37 o tan37 o National 4 REL 1.3a REL 1.3b

9. www.mathsrevision.com Using a calculator to find tan ratio What Goes In The Box ? Find the tan ratio to 3 decimal places. tan 30 o tan 0 o tan 45 o tan 60 o tan 20 o tan 70 o = 0.577= 0.000 = 1.000 = 1.732= 0.364= 2.747 National 4 REL 1.3a REL 1.3b

10. www.mathsrevision.com Trigonometry Adj x°x°x°x° Tan x° = O p p o s i t e Opp Adjacent National 4 REL 1.3a REL 1.3b

11. www.mathsrevision.com Trigonometry National 4 REL 1.3a REL 1.3b 60° 12 m Adjacent Opposite hypotenuse Copy this!

12. www.mathsrevision.com Trigonometry National 4 REL 1.3a REL 1.3b Tan x° = Opp Adj Tan 60° = Opp 12 = Opp12 x Tan 60° Opp =12 x Tan 60°= 20.8m (1 d.p.) Copy this!

13. www.mathsrevision.com Trigonometry National 4 REL 1.3a REL 1.3b Tan Ratio Ex1 Ch16 Page 198

14. Starter Questions Starter Questions www.mathsrevision.com xoxo National 4 REL 1.3a REL 1.3b

15. 18-May-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Learning Intention Success Criteria 2.Use tan of an angle to solve REAL LIFE problems. 1.To use tan of the angle to solve REAL LIFE problems. Angles & Triangles 1.Write down tan ratio. National 4 REL 1.3a REL 1.3b

16. www.mathsrevision.com Trigonometry Adj x°x°x°x° Tan x° = O p p o s i t e Opp Adjacent National 4 REL 1.3a REL 1.3b

17. www.mathsrevision.com Trigonometry Example 65° Tan x° = Opp Adj Hyp h 8m Tan 65° = h 8 = h8 x Tan 65° h =8 x Tan 65°= 17.2m (1 d.p.) Adj National 4 REL 1.3 a + 1.3b Find the height h SOH CAH TOASOH CAH TOA

18. www.mathsrevision.com Trigonometry National 4 REL 1.3a REL 1.3b SOH CAH TOASOH CAH TOA 18-May-15Compiled by Mr. Lafferty Maths Dept. Use the tan ratio to find the height h of the tree to 2 decimal places. 47 o 8m rod

19. www.mathsrevision.com Trigonometry National 4 REL 1.3a REL 1.3b 6o6o 18-May-15 Compiled by Mr. Lafferty Maths Dept. Aeroplane a = 15 c Lennoxtown Airport Q1.An aeroplane is preparing to land at Glasgow Airport. It is over Lennoxtown at present which is 15km from the airport. The angle of descent is 6 o. What is the height of the plane ? Example 2 SOH CAH TOASOH CAH TOA

20. www.mathsrevision.com Trigonometry National 4 REL 1.3a REL 1.3b 18-May-15Compiled by Mr. Lafferty Maths Dept. Use the tan ratio to calculate how far the ladder is away from the building. 45 o 12m ladder d m SOH CAH TOASOH CAH TOA

21. www.mathsrevision.com Trigonometry National 4 REL 1.3a REL 1.3b 6o6o 18-May-15Compiled by Mr. Lafferty Maths Dept. Aeroplane a = 1.58 km Lennoxtown Airport Q1.An aeroplane is preparing to land at Glasgow Airport. It is over Lennoxtown at present. It is at a height of 1.58 km above the ground. It ‘s angle of descent is 6 o. How far is it from the airport to Lennoxtown? Example 2 SOH CAH TOASOH CAH TOA

22. www.mathsrevision.com Trigonometry National 4 REL 1.3a REL 1.3b Now try Exercise 2 Ch16 (page 199)

23. Starter Questions Starter Questions www.mathsrevision.com 10cm P Q R 6cm 7cm 10cm National 4 REL 1.3a REL 1.3b

24. 18-May-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Learning Intention Success Criteria 2.Use tan of an angle to solve problems. 1.To use tan of the angle to solve problems. Angles & Triangles 1.Write down tan ratio. National 4 REL 1.3a REL 1.3b

25. Using Tan to calculate angles www.mathsrevision.com National 4 REL 1.3a REL 1.3b

26. www.mathsrevision.com Using a calculator to find angle given tan ratio What Goes In The Box ? Find the angle given the tan ratio to 1 decimal places. tan -1 1 tan -1 0.5tan -1 0.8 tan -1 0.6tan -1 2tan -1 3 = 45.0 o = 26.6 o = 38.7 o = 31.0 o = 64.4 o = 71.6 o National 4 REL 1.3a REL 1.3b

27. www.mathsrevision.com Trigonometry 18 12 Example x°x° Tan x° = Opp Adj Hyp SOH CAH TOASOH CAH TOA 12m Tan x° = = 1.5Tan x° Adj 18m Calculate the tan x o ratio Q P R National 4 REL 1.3a REL 1.3b

28. www.mathsrevision.com Trigonometry National 4 REL 1.3a REL 1.3b = 1.5Tan x° How do we find x°? We need to use Tan ⁻ ¹ on the calculator. 2 nd Tan ⁻ ¹is written above Tan Tan ⁻ ¹ To get this press Tan Followed by Calculate the size of angle x o

29. www.mathsrevision.com Trigonometry National 4 REL 1.3a REL 1.3b x = Tan ⁻ ¹ 1.5 = 56.3° (1 d.p.) = 1.5Tan x° 2 nd Tan Tan ⁻ ¹ Press Enter = 1.5

30. www.mathsrevision.com Trigonometry National 4 REL 1.3a REL 1.3b Process 1.Identify Hyp, Opp and Adj 2.Write down ratio Tan x o = Opp Adj 3.Calculate x o 2 nd Tan Tan ⁻ ¹

31. www.mathsrevision.com Trigonometry National 4 REL 1.3a REL 1.3b 18-May-15Compiled by Mr. Lafferty Maths Dept. Use the tan ratio to calculate the angle that the support wire makes with the ground. xoxo 11m 4 m SOH CAH TOASOH CAH TOA

32. www.mathsrevision.com Trigonometry National 4 REL 1.3a REL 1.3b Now try Exercise 3 Ch16 Page 202

33. Starter Questions Starter Questions 18-May-15Created by Mr.Lafferty Maths Dept www.mathsrevision.com National 4 REL 1.3a REL 1.3b

34. 18-May-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Learning Intention Success Criteria 2.Use sine ratio to find an angle. 1.Definite the sine ratio and show how to find an angle using this ratio. Angles & Triangles 1.Write down sine ratio. National 4 REL 1.3a REL 1.3b

35. www.mathsrevision.com Trigonometry The Sine Ratio x°x°x°x° Sin x° = O p p o s i t e Opp Hyp h y p o t e n u s e National 4 REL 1.3a REL 1.3b

36. www.mathsrevision.com Using a calculator to find sin ratio What Goes In The Box ? Find the sin ratio to 3 decimal places. sin 30 o sin 0 o sin 45 o sin 60 o sin 20 o sin 90 o = 0.500= 0.000 = 0.707 = 0.866= 0.342= 1.000 National 4 REL 1.3a REL 1.3b

37. www.mathsrevision.com Trigonometry National 4 REL 1.3a REL 1.3b Example 34° Sin x° = Opp Hyp h 11cm Sin 34° = h 11 = h 11 x Sin 34° h =11 x Sin 34°= 6.2cm (1 d.p.) Find the height h SOH CAH TOASOH CAH TOA

38. www.mathsrevision.com Trigonometry National 4 REL 1.3a REL 1.3b SOH CAH TOASOH CAH TOA 18-May-15Compiled by Mr. Lafferty Maths Dept. The support rope is 11.7m long. The angle between the rope and ground is 70 o. Use the sine ratio to calculate the height of the flag pole. 70 o h 11.7m

39. www.mathsrevision.com Trigonometry SOH CAH TOASOH CAH TOA Example 72° Sin x° = Opp Hyp Sin 72° = 5 r r = 5.3 km 5km AB C r A road AB is right angled at B. The road BC is 5 km. Calculate the length of the new road AC. National 4 REL 1.3a REL 1.3b

40. www.mathsrevision.com Trigonometry National 4 REL 1.3a REL 1.3b Now try Extension Booklet Ex4E (page 83)

41. Starter Questions Starter Questions www.mathsrevision.com National 4 REL 1.3a REL 1.3b

42. 18-May-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Learning Intention Success Criteria 2.Use sine ratio to find angle REAL-LIFE problems. 1.To show how to use the sine ratio to find angle in REAL-LIFE problems. Angles & Triangles 1.Write down sine ratio. National 4 REL 1.3a REL 1.3b

43. Using Sin to calculate angles www.mathsrevision.com National 4 REL 1.3a REL 1.3b

44. www.mathsrevision.com Using a calculator to find angle given sin ratio What Goes In The Box ? Find the angle given the sin ratio to 1 decimal places. sin -1 1 sin -1 0.5sin -1 0.8 sin -1 0.6sin -1 0.1sin -1 0.9 = 90.0 o = 30.0 o = 53.1 o = 36.9 o = 5.7 o = 64.2 o National 4 REL 1.3a REL 1.3b

45. www.mathsrevision.com Trigonometry Example x°x°x°x° Sin x° = Opp Hyp 6m 9m Sin x° = 6 9 = 0.667 (3 d.p.)Sin x° Find the x o SOH CAH TOASOH CAH TOA National 4 REL 1.3a REL 1.3b

46. www.mathsrevision.com Trigonometry National 4 REL 1.3a REL 1.3b =0.667 (3 d.p.)Sin x° How do we find x°? We need to use Sin ⁻ ¹ on the calculator. 2 nd Sin ⁻ ¹is written above Sin Sin ⁻ ¹ To get this press Sin Followed by

47. www.mathsrevision.com Trigonometry National 4 REL 1.3a REL 1.3b x = Sin ⁻ ¹ 0.667 = 41.8° (1 d.p.) = 0.667 (3 d.p.)Sin x° 2 nd Sin Sin ⁻ ¹ Press Enter = 0.667

48. www.mathsrevision.com Trigonometry National 4 REL 1.3a REL 1.3b SOH CAH TOASOH CAH TOA 18-May-15Compiled by Mr. Lafferty Maths Dept. Use the sine ratio to find the angle of the ramp. xoxo 10m 20 m

49. www.mathsrevision.com Trigonometry National 4 REL 1.3a REL 1.3b Now try Extension Booklet Ex5E (page 85)

50. Starter Questions Starter Questions www.mathsrevision.com National 4 REL 1.3a REL 1.3b

51. 18-May-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Learning Intention Success Criteria 2.Use cosine ratio to find a length or angle. 1.Definite the cosine ratio and show how to find an length or angle using this ratio. Angles & Triangles 1.Write down cosine ratio. National 4 REL 1.3a REL 1.3b

52. www.mathsrevision.com Using a calculator to find cos ratio What Goes In The Box ? Find the cos ratio to 3 decimal places. cos 30 o cos 0 o cos 45 o cos 60 o cos 20 o cos 90 o = 0.866= 1.000 = 0.707 = 0.500= 0.940= 0.000 National 4 REL 1.3a REL 1.3b

53. www.mathsrevision.com Trigonometry The Cosine Ratio Cos x° = Adjacent Adj x°x°x°x° Hyp h y p o t e n u s e National 4 REL 1.3a REL 1.3b

54. www.mathsrevision.com Trigonometry National 4 REL 1.3a REL 1.3b SOH CAH TOASOH CAH TOA Example 40° Cos x° = Opp Adj Hyp b 35mm Cos 40° = b 35 = b35 x Cos 40° b =35 x Cos 40°= 26.8mm (1 d.p.) Adj Find the adjacent length b

55. www.mathsrevision.com Trigonometry National 4 REL 1.3a REL 1.3b Now try Extension Booklet Ex6E (page 88 Q1 to Q5)

56. 18-May-15Created by Mr.Lafferty Maths Dept Starter Questions Starter Questions www.mathsrevision.com National 4 REL 1.3a REL 1.3b

57. 18-May-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Learning Intention Success Criteria 2.Use cosine ratio to find a length or angle. 1.Definite the cosine ratio and show how to find an length or angle using this ratio. Angles & Triangles 1.Write down cosine ratio. National 4 REL 1.3a REL 1.3b

58. Using Cos to calculate angles www.mathsrevision.com National 4 REL 1.3a REL 1.3b

59. www.mathsrevision.com Using a calculator to find angle given cos ratio What Goes In The Box ? Find the angle given the cos ratio to 1 decimal places. cos -1 1 cos -1 0cos -1 0.8 cos -1 0.6cos -1 0.1cos -1 0.9 = 0.0 o = 90.0 o = 36.9 o = 53.1 o = 84.3 o = 25.8 o National 4 REL 1.3a REL 1.3b

60. www.mathsrevision.com Trigonometry S O H C A H T O A Example x°x°x°x° Cos x° = Opp Adj Hyp 45cm Cos x° = 34 45 = 0.756 (3 d.p.)Cos x° x = Cos ⁻ ¹0.756 =41° Adj 34cm Find the angle x o National 4 REL 1.3a REL 1.3b

61. www.mathsrevision.com Trigonometry National 4 REL 1.3a REL 1.3b Now try Extension Booklet Ex6E (page 89 Q6 to Q8)

62. Starter Questions Starter Questions www.mathsrevision.com xoxo 6 8 10 National 4 REL 1.3a REL 1.3b

63. The Three Ratios www.mathsrevision.com Cosine Sine Tangent Sine Tangent Cosine Sine www.mathsrevision.com opposite adjacent hypotenuse National 4 REL 1.3a REL 1.3b

64. www.mathsrevision.com Trigonometry Sin x° = Opp Hyp Cos x° = Adj Hyp Tan x° = Opp Adj C A HT O AS O H National 4 REL 1.3a REL 1.3b

65. www.mathsrevision.com Trigonometry S O H C A H T O A Copy this! 1. Write down Process Identify what you want to find what you know 3. 2. National 4 REL 1.3a REL 1.3b

66. www.mathsrevision.com Trigonometry Past Paper Type Questions S O H C A H T O A National 4 REL 1.3a REL 1.3b

67. www.mathsrevision.com Trigonometry Past Paper Type Questions (4 marks) S O H C A H T O A National 4 REL 1.3a REL 1.3b

68. www.mathsrevision.com Trigonometry Past Paper Type Questions S O H C A H T O A National 4 REL 1.3a REL 1.3b

69. www.mathsrevision.com Trigonometry Past Paper Type Questions S O H C A H T O A 4 marks National 4 REL 1.3a REL 1.3b

70. www.mathsrevision.com Trigonometry Past Paper Type Questions S O H C A H T O A National 4 REL 1.3a REL 1.3b

71. www.mathsrevision.com Trigonometry Past Paper Type Questions (4marks) S O H C A H T O A National 4 REL 1.3a REL 1.3b

72. www.mathsrevision.com Trigonometry Past Paper Type Questions S O H C A H T O A National 4 REL 1.3a REL 1.3b

73. www.mathsrevision.com Trigonometry Past Paper Type Questions (4marks) S O H C A H T O A National 4 REL 1.3a REL 1.3b

74. www.mathsrevision.com Trigonometry National 4 REL 1.3a REL 1.3b Now try Extension Booklet Ex7E (page 91)