2. The Tangent Ratio The Tangent using Angle The Sine of an Angle The Sine Ration In Action The Cosine of an Angle Mixed Problems The Tangent Ratio in Action The Tangent (The Adjacent side) The Tangent (Finding Angle) The Sine ( Finding the Hypotenuse)

3. 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.

4. Trigonometry means “triangle” and “measurement”. Adjacent Opposite x°x°x°x° hypotenuse We will be using right-angled triangles.

5. 30° Adjacent Opposite hypotenuse Opposite Adjacent = 0.6 Mathemagic!

6. 45° Adjacent Opposite hypotenuse Opposite Adjacent = 1 Try another!

7. For an angle of 30°, Opposite Adjacent = 0.6 We write tan 30° = 0.6 Opposite Adjacent is called the tangent of an angle.

8. Tan 25° 0.466 Tan 26° 0.488 Tan 27° 0.510 Tan 28° 0.532 Tan 29° 0.554 Tan 30° 0.577 Tan 31° 0.601 Tan 32° 0.625 Tan 33° 0.649 Tan 34° 0.675 Tan 30° = 0.577 Accurate to 3 decimal places! The ancient Greeks discovered this and repeated this for all possible angles.

9. Now-a-days we can use calculators instead of tables to find the Tan of an angle. Tan On your calculator press Notice that your calculator is incredibly accurate!! Followed by 30, and press = Accurate to 9 decimal places!

10. What’s the point of all this???Don’t worry, you’re about to find out!

11. 12 m How high is the tower? Opp 60°

12. 12 m Adjacent Opposite hypotenuse Copy this!

13. Tan x° = Opp Adj Tan 60° = Opp 12 = Opp12 x Tan 60° Opp =12 x Tan 60°= 20.8m (1 d.p.) Copy this!

14. So the tower’s 20.8 m high! Don’t worry, you’ll be trying plenty of examples!! 20.8m

15. Adj x°x°x°x° Tan x° = O p p o s i t e Opp Adjacent

16. 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 Find the height h SOH CAH TOASOH CAH TOA

17. 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.

18. Using Tan to calculate angles

19. 18 12 Example x°x°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

20. = 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

21. x = Tan ⁻ ¹ 1.5 = 56.3° (1 d.p.) = 1.5Tan x° 2 nd Tan Tan ⁻ ¹ Press Enter = 1.5

22. 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

23. 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.

24. SOH CAH TOASOH CAH TOA Use the tan ratio to find the height h of the tree to 2 decimal places. 47 o 8m rod

25. 6o6o 20-Oct-15 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

26. Learning Intention Success Criteria 2.Use tan of an angle to solve find adjacent length. 1.To use tan of the angle to find adjacent length. Angles & Triangles 1.Write down tan ratio.

27. 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

28. 6o6o 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

29. Learning Intention Success Criteria 2.Use tan ratio to find an angle. 1.To show how to find an angle using tan ratio. Angles & Triangles 1.Write down tan ratio.

30. 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

31. Use the tan ratio to find the angle of take-off. xoxo 88m 500 m SOH CAH TOASOH CAH TOA

32. 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.

33. 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

34. 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

35. Using Sin to calculate angles

36. 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

37. =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

38. 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

39. Learning Intention Success Criteria 2.Use sine ratio to solve REAL-LIFE problems. 1.To show how to use the sine ratio to solve REAL-LIFE problems. Angles & Triangles 1.Write down sine ratio.

40. SOH CAH TOASOH CAH TOA 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

41. SOH CAH TOASOH CAH TOA Use the sine ratio to find the angle of the ramp. xoxo 10m 20 m

42. Learning Intention Success Criteria 2.Use sine ratio to find the hypotenuse. 1.To show how to calculate the hypotenuse using the sine ratio. Angles & Triangles 1.Write down sine ratio.

43. 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.

44. 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.

45. The Cosine Ratio Cos x° = Adjacent Adj x°x°x°x° Hyp h y p o t e n u s e

46. 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

47. Using Cos to calculate angles

48. 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

49. The Three Ratios Cosine Sine Tangent Sine Tangent Cosine Sine opposite adjacent hypotenuse

50. Sin x° = Opp Hyp Cos x° = Adj Hyp Tan x° = Opp Adj C A HT O AS O H

51. 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.

52. Past Paper Type Questions S O H C A H T O A

53. Past Paper Type Questions (4 marks) S O H C A H T O A

54. Past Paper Type Questions S O H C A H T O A

55. Past Paper Type Questions S O H C A H T O A 4 marks

56. Past Paper Type Questions S O H C A H T O A

57. Past Paper Type Questions (4marks) S O H C A H T O A

58. Past Paper Type Questions S O H C A H T O A

59. Past Paper Type Questions (4marks) S O H C A H T O A