1. 13-Aug-15Created by Mr. Lafferty Maths Dept. Trigonometry www.mathsrevision.com Cosine Rule Finding a Length Sine Rule Finding a length Mixed Problems Nat 5 Sine Rule Finding an Angle Cosine Rule Finding an Angle Area of ANY Triangle Revision (S O H)(C A H)(T O A) Exam Type Questions

2. Starter Questions Starter Questions www.mathsrevision.com xoxo 6 8 10 Nat 5

3. 13-Aug-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Learning Intention Success Criteria 2.Use inding an angle or length given a right-angled triangle. 2.Use S O HC A HT O A to finding an angle or length given a right-angled triangle. 1.We are revising S O HC A HT O A process. Angles & Triangles 1.Know the tree ratios for 1.Know the tree ratios for S O HC A HT O A. Nat 5

4. The Three Ratios www.mathsrevision.com Cosine Sine Tangent Sine Tangent Cosine Sine www.mathsrevision.com opposite adjacent hypotenuse Nat 5

5. www.mathsrevision.com Trigonometry Sin x° = Opp Hyp Cos x° = Adj Hyp Tan x° = Opp Adj C A HT O AS O H Nat 5

6. 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. Nat 5

7. (4 marks) S O H C A H T O A

10. (4marks) S O H C A H T O A

11. www.mathsrevision.com Trigonometry Nat 5 Now try N5 TJ Ex8.1 Q3 onwards Ch8 (page 71)

12. 13-Aug-15Created by Mr. Lafferty Maths Dept. Starter Questions Starter Questions www.mathsrevision.com Nat 5

13. 13-Aug-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Learning Intention Success Criteria 1.Know the formula for the area of any triangle. 1. We are learning how to apply the Area formula for ANY triangle. Nat 5 Area of ANY Triangle 2.Use formula to find area of any triangle given two length and angle in between.

14. 13-Aug-15Created by Mr Lafferty Maths Dept Labelling Triangles www.mathsrevision.com Nat 5 A B C A a B b C c Small letters a, b, c refer to distances Capital letters A, B, C refer to angles In Mathematics we have a convention for labelling triangles.

15. F E D F E D 13-Aug-15Created by Mr Lafferty Maths Dept Labelling Triangles www.mathsrevision.com Nat 5 d e f Have a go at labelling the following triangle.

16. General Formula for Area of ANY Triangle Consider the triangle below: AoAo BoBo CoCo a b c h Area = ½ x base x height What does the sine of A o equal Change the subject to h. h = b sinA o Substitute into the area formula www.mathsrevision.com Nat 5

17. 13-Aug-15Created by Mr Lafferty Maths Dept Area of ANY Triangle www.mathsrevision.com Nat 5 A B C A a B b C c The area of ANY triangle can be found by the following formula. Another version Another version Key feature To find the area you need to know 2 sides and the angle in between (SAS) Demo

18. 13-Aug-15Created by Mr Lafferty Maths Dept Area of ANY Triangle www.mathsrevision.com Nat 5 A B C A 20cm B 25cm C c Example : Find the area of the triangle. The version we use is 30 o

19. 13-Aug-15Created by Mr Lafferty Maths Dept Area of ANY Triangle www.mathsrevision.com Nat 5 D E F 10cm 8cm Example : Find the area of the triangle. The version we use is 60 o

20. What Goes In The Box ? Calculate the areas of the triangles below: (1) 23 o 15cm 12.6cm (2) 71 o 5.7m 6.2m A = 36.9cm 2 A = 16.7m 2 www.mathsrevision.com Nat 5 Key feature Remember (SAS)

21. www.mathsrevision.com Trigonometry Nat 5 Now try N5 TJ Ex 8.2 Ch8 (page 73)

22. 13-Aug-15Created by Mr. Lafferty Maths Dept. Starter Questions Starter Questions www.mathsrevision.com Nat 5

23. 13-Aug-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Learning Intention Success Criteria 1.Know how to use the sine rule to solve REAL LIFE problems involving lengths showing ALL appropriate working. 1. We are learning how to use the sine rule to solve REAL LIFE problems involving finding the length of a side of a triangle. Sine Rule Nat 5

24. C B A 13-Aug-15Created by Mr Lafferty Maths Dept Sine Rule www.mathsrevision.com Nat 5 a b c The Sine Rule can be used with ANY triangle as long as we have been given enough information. Works for any Triangle Demo

25. Deriving the rule B C A b c a Consider a general triangle ABC. The Sine Rule Draw CP perpendicular to BA P This can be extended to or equivalently

26. Calculating Sides Using The Sine Rule 10m 34 o 41 o a Match up corresponding sides and angles: Rearrange and solve for a. Example 1 : Find the length of a in this triangle. www.mathsrevision.com Nat 5 A B C Demo

27. Calculating Sides Using The Sine Rule www.mathsrevision.com Nat 5 10m 133 o 37 o d = 12.14m Match up corresponding sides and angles: Rearrange and solve for d. Example 2 : Find the length of d in this triangle. C D E Demo

28. What goes in the Box ? Find the unknown side in each of the triangles below: (1) 12cm 72 o 32 o a (2) 93 o b 47 o 16mm A = 6.7cm B = 21.8mm www.mathsrevision.com Nat 5 13-Aug-15Created by Mr Lafferty Maths Dept

29. www.mathsrevision.com Trigonometry Nat 5 Now try N5 TJ Ex 8.3 Ch8 (page 76)

30. 13-Aug-15Created by Mr. Lafferty Maths Dept. Starter Questions Starter Questions www.mathsrevision.com Nat 5

31. 13-Aug-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Learning Intention Success Criteria 1.Know how to use the sine rule to solve problems involving angles. 1. We are learning how to use the sine rule to solve problems involving finding an angle of a triangle. Sine Rule Nat 5

32. Calculating Angles Using The Sine Rule Example 1 : Find the angle A o A 45m 23 o 38m Match up corresponding sides and angles: Rearrange and solve for sin A o = 0.463 Use sin -1 0.463 to find A o www.mathsrevision.com Nat 5 B C Demo

33. Calculating Angles Using The Sine Rule 143 o 75m 38m X = 0.305 Example 2 : Find the angle X o Match up corresponding sides and angles: Rearrange and solve for sin X o Use sin -1 0.305 to find X o www.mathsrevision.com Nat 5 Y Z Demo

34. What Goes In The Box ? Calculate the unknown angle in the following: (1) 14.5m 8.9m AoAo 100 o (2) 14.7cm BoBo 14 o 12.9cm A o = 37.2 o B o = 16 o www.mathsrevision.com Nat 5

35. www.mathsrevision.com Trigonometry Nat 5 Now try N5 TJ Ex 8.4 Ch8 (page 79)

36. 13-Aug-15Created by Mr. Lafferty Maths Dept. Starter Questions Starter Questions www.mathsrevision.com Nat 5

37. 13-Aug-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Learning Intention Success Criteria 1.Know when to use the cosine rule to solve problems. 1. We are learning when to use the cosine rule to solve problems involving finding the length of a side of a triangle. Cosine Rule Nat 5 2. Solve problems that involve finding the length of a side.

38. C B A 13-Aug-15Created by Mr Lafferty Maths Dept Cosine Rule www.mathsrevision.com Nat 5 a b c The Cosine Rule can be used with ANY triangle as long as we have been given enough information. Works for any Triangle

39. Deriving the rule A B C a b c Consider a general triangle ABC. We require a in terms of b, c and A. Draw BP perpendicular to AC b P x b - x BP 2 = a 2 – (b – x) 2 Also: BP 2 = c 2 – x 2  a 2 – (b – x) 2 = c 2 – x 2  a 2 – (b 2 – 2bx + x 2 ) = c 2 – x 2  a 2 – b 2 + 2bx – x 2 = c 2 – x 2  a 2 = b 2 + c 2 – 2bx*  a 2 = b 2 + c 2 – 2bcCosA *Since Cos A = x/c  x = cCosA When A = 90 o, CosA = 0 and reduces to a 2 = b 2 + c 2 1 When A > 90 o, CosA is negative,  a 2 > b 2 + c 2 2 When A b 2 + c 2 3 The Cosine Rule The Cosine Rule generalises Pythagoras’ Theorem and takes care of the 3 possible cases for Angle A. a 2 > b 2 + c 2 a 2 < b 2 + c 2 a 2 = b 2 + c 2 A A A 1 2 3 Pythagoras + a bit Pythagoras - a bit Pythagoras

40. a 2 = b 2 + c 2 – 2bcCosA Applying the same method as earlier to the other sides produce similar formulae for b and c. namely: b 2 = a 2 + c 2 – 2acCosB c 2 = a 2 + b 2 – 2abCosC A B C a b c The Cosine Rule The Cosine rule can be used to find: 1. An unknown side when two sides of the triangle and the included angle are given (SAS). 2. An unknown angle when 3 sides are given (SSS). Finding an unknown side.

41. 13-Aug-15Created by Mr Lafferty Maths Dept Cosine Rule www.mathsrevision.com Nat 5 How to determine when to use the Cosine Rule. Works for any Triangle 1. Do you know ALL the lengths. 2. Do you know 2 sides and the angle in between. SAS OR If YES to any of the questions then Cosine Rule Otherwise use the Sine Rule Two questions

42. Using The Cosine Rule Example 1 : Find the unknown side in the triangle below: L 5m 12m 43 o Identify sides a,b,c and angle A o a =Lb =5c =12A o =43 o Write down the Cosine Rule. a 2 =b2b2 +c2c2 -2bccosA o Substitute values to find a 2. a 2 =5252 +12 2 - 2 x 5 x 12 cos 43 o a 2 =25 + 144-(120 x0.731 ) a 2 =81.28 Square root to find “a”. a = L = 9.02m www.mathsrevision.com Nat 5 Works for any Triangle Demo

43. Example 2 : Find the length of side M. 137 o 17.5 m 12.2 m M Identify the sides and angle. a = Mb = 12.2C = 17.5A o = 137 o Write down Cosine Rule a 2 =b2b2 +c2c2 -2bccosA o a 2 = 12.2 2 + 17.5 2 – ( 2 x 12.2 x 17.5 x cos 137 o ) a 2 = 148.84 + 306.25 – ( 427 x – 0.731 ) Notice the two negative signs. a 2 = 455.09 + 312.137 a 2 = 767.227 a = M = 27.7m Using The Cosine Rule www.mathsrevision.com Nat 5 Works for any Triangle Demo

44. What Goes In The Box ? Find the length of the unknown side in the triangles: (1) 78 o 43cm 31cm L (2) 8m 5.2m 38 o M L = 47.5cm M =5.05m www.mathsrevision.com Nat 5

45. www.mathsrevision.com Trigonometry Nat 5 Now try N5 TJ Ex 8.5 Ch8 (page 81)

46. 13-Aug-15Created by Mr. Lafferty Maths Dept. Starter Questions Starter Questions www.mathsrevision.com Nat 5 54 o

47. 13-Aug-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Learning Intention Success Criteria 1.Know when to use the cosine rule to solve problems. 1.Know when to use the cosine rule to solve REAL LIFE problems. 1. We are learning when to use the cosine rule to solve REAL LIFE problems involving finding an angle of a triangle. Cosine Rule Nat 5 2. Solve problems that involve finding an angle of a triangle. 2. Solve REAL LIFE problems that involve finding an angle of a triangle.

48. C B A 13-Aug-15Created by Mr Lafferty Maths Dept Cosine Rule www.mathsrevision.com Nat 5 a b c The Cosine Rule can be used with ANY triangle as long as we have been given enough information. Works for any Triangle

49. Finding Angles Using The Cosine Rule Consider the Cosine Rule again: a 2 =b2b2 +c2c2 -2bccosA o We are going to change the subject of the formula to cos A o Turn the formula around: b 2 + c 2 – 2bc cos A o = a 2 Take b 2 and c 2 across. -2bc cos A o = a 2 – b 2 – c 2 Divide by – 2 bc. Divide top and bottom by -1 You now have a formula for finding an angle if you know all three sides of the triangle. www.mathsrevision.com Nat 5 Works for any Triangle

50. AoAo 16cm 9cm11cm Write down the formula for cos A o Label and identify A o and a, b and c. A o = ? a = 11b = 9c = 16 Substitute values into the formula. Calculate cos A o. Cos A o =0.75 Use cos -1 0.75 to find A o A o = 41.4 o Example 1 : Calculate the unknown angle A o. Finding Angles Using The Cosine Rule www.mathsrevision.com Nat 5 Works for any Triangle Demo

51. Example 2: Find the unknown Angle y o in the triangle: 26cm 15cm 13cm yoyo Write down the formula. Identify the sides and angle. A o = y o a = 26b = 15c = 13 Find the value of cosA o cosA o = - 0.723 The negative tells you the angle is obtuse. A o = y o =136.3 o www.mathsrevision.com Nat 5 Finding Angles Using The Cosine Rule Works for any Triangle Demo

52. What Goes In The Box ? Calculate the unknown angles in the triangles below: (1) 10m 7m 5m AoAo BoBo (2) 12.7cm 7.9cm 8.3cm A o =111.8 o B o = 37.3 o www.mathsrevision.com Nat 5

53. www.mathsrevision.com Trigonometry Nat 5 Now try N5 TJ Ex 8.6 Ch8 (page 84)

54. 13-Aug-15Created by Mr. Lafferty Maths Dept. Starter Questions Starter Questions www.mathsrevision.com Nat 5 61 o

55. 13-Aug-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Learning Intention Success Criteria 1.Be able to recognise the correct trigonometric formula to use to solve a problem involving triangles. 1. We are learning to use our knowledge gained so far to solve various trigonometry problems. Mixed problems Nat 5

56. SOH CAH TOASOH CAH TOA 25 o 15 m A D The angle of elevation of the top of a building measured from point A is 25 o. At point D which is 15m closer to the building, the angle of elevation is 35 o Calculate the height of the building. T B Angle TDA = 145 o Angle DTA = 10 o 35 o 36.5 180 – 35 = 145 o 180 – 170 = 10 o Exam Type Questions

57. A fishing boat leaves a harbour (H) and travels due East for 40 miles to a marker buoy (B). At B the boat turns left and sails for 24 miles to a lighthouse (L). It then returns to harbour, a distance of 57 miles. (a)Make a sketch of the journey. (b)Find the bearing of the lighthouse from the harbour. (nearest degree) H 40 miles 24 miles B L 57 miles A Exam Type Questions

58. A The angle of elevation of the top of a column measured from point A, is 20 o. The angle of elevation of the top of the statue is 25 o. Find the height of the statue when the measurements are taken 50 m from its base 50 m Angle BCA = 70 o Angle ACT = Angle ATC = 110 o 65 o 53.21 m B T C 180 – 110 = 70 o 180 – 70 = 110 o 180 – 115 = 65 o 20 o 25 o 5o5o SOH CAH TOASOH CAH TOA Exam Type Questions

59. An AWACS aircraft takes off from RAF Waddington (W) on a navigation exercise. It flies 530 miles North to a point (P) as shown, It then turns left and flies to a point (Q), 670 miles away. Finally it flies back to base, a distance of 520 miles. Find the bearing of Q from point P. P 670 miles W 530 miles Not to Scale Q 520 miles Exam Type Questions

60. www.mathsrevision.com Trigonometry Nat 5 Now try N5 TJ Ex 8.7 & 8.8 Ch8 (page 85)