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Trigonometry

S4 Credit. Trigonometry. Sine Rule Finding a length. Sine Rule Finding an Angle. Cosine Rule Finding a Length. Cosine Rule Finding an Angle. www.mathsrevision.com. Area of ANY Triangle. Mixed Problems. S4 Credit. Starter Questions. www.mathsrevision.com. Sine Rule. S4 Credit.

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Trigonometry

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  1. S4 Credit Trigonometry Sine Rule Finding a length Sine Rule Finding an Angle Cosine Rule Finding a Length Cosine Rule Finding an Angle www.mathsrevision.com Area of ANY Triangle Mixed Problems Created by Mr. Lafferty Maths Dept.

  2. S4 Credit Starter Questions www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

  3. Sine Rule S4 Credit Learning Intention Success Criteria • Know how to use the sine rule to solve REAL LIFE problems involving lengths. • 1. To show how to use the sine rule to solve REAL LIFE problems involving finding the length of a side of a triangle . www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

  4. Works for any Triangle S4 Credit Sine Rule The Sine Rule can be used with ANY triangle as long as we have been given enough information. B a www.mathsrevision.com c C b A Created by Mr Lafferty Maths Dept

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

  6. a 10m 34o 41o Calculating Sides Using The Sine Rule S4 Credit Example 1 : Find the length of a in this triangle. B C A Match up corresponding sides and angles: www.mathsrevision.com Rearrange and solve for a.

  7. 10m 133o 37o d Calculating Sides Using The Sine Rule S4 Credit Example 2 : Find the length of d in this triangle. D E C Match up corresponding sides and angles: www.mathsrevision.com Rearrange and solve for d. = 12.14m

  8. 12cm (1) (2) b 47o 32o a 72o 16mm 93o What goes in the Box ? S4 Credit Find the unknown side in each of the triangles below: www.mathsrevision.com A = 6.7cm B = 21.8mm Created by Mr Lafferty Maths Dept

  9. Sine Rule S4 Credit Now try MIA Ex 2.1 Ch12 (page 247) www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

  10. S4 Credit Starter Questions www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

  11. Sine Rule S4 Credit Learning Intention Success Criteria • Know how to use the sine rule to solve problems involving angles. • 1. To show how to use the sine rule to solve problems involving finding an angle of a triangle . www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

  12. 45m 38m 23o A Calculating Angles Using The Sine Rule S4 Credit B Example 1 : Find the angle Ao C Match up corresponding sides and angles: www.mathsrevision.com Rearrange and solve for sin Ao Use sin-1 0.463 to find Ao = 0.463

  13. 75m X 38m 143o Calculating Angles Using The Sine Rule S4 Credit Example 2 : Find the angle Xo Z Y Match up corresponding sides and angles: www.mathsrevision.com Rearrange and solve for sin Xo Use sin-1 0.305 to find Xo = 0.305

  14. (1) 8.9m 100o (2) Ao Bo 12.9cm 14.5m 14o 14.7cm What Goes In The Box ? S4 Credit Calculate the unknown angle in the following: www.mathsrevision.com Ao = 37.2o Bo = 16o

  15. Sine Rule S4 Credit Now try MIA Ex3.1 Ch12 (page 249) www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

  16. S4 Credit Starter Questions www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

  17. Cosine Rule S4 Credit Learning Intention Success Criteria • Know when to use the cosine rule to solve problems. • 1. To show when to use the cosine rule to solve problems involving finding the length of a side of a triangle . • 2. Solve problems that involve finding the length of a side. www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

  18. Works for any Triangle S4 Credit Cosine Rule The Cosine Rule can be used with ANY triangle as long as we have been given enough information. B a www.mathsrevision.com c C b A Created by Mr Lafferty Maths Dept

  19. The Cosine Rule The Cosine Rule generalises Pythagoras’ Theorem and takes care of the 3 possible cases for Angle A. A Consider a general triangle ABC. We require a in terms of b, c and A. B a2 = b2 + c2 a c A P C A x b - x a2 > b2 + c2 b A When A = 90o, CosA = 0 and reduces to a2 = b2 + c2 1 1 b When A > 90o, CosA is negative,  a2 > b2 + c2 2 2 a2 < b2 + c2 When A < 90o, CosA is positive,  a2 > b2 + c2 3 3 Deriving the rule • BP2 = a2 – (b – x)2 • Also: BP2 = c2 – x2 • a2 – (b – x)2 = c2 – x2 • a2 – (b2 – 2bx + x2) = c2 – x2 • a2 – b2 + 2bx – x2 = c2 – x2 • a2 = b2 + c2 – 2bx* • a2 = b2 + c2 – 2bcCosA Draw BP perpendicular to AC *Since Cos A = x/c  x = cCosA Pythagoras Pythagoras + a bit Pythagoras - a bit

  20. 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). B a c C A b Finding an unknown side. a2 = b2 + c2 – 2bcCosA Applying the same method as earlier to the other sides produce similar formulae for b and c. namely: b2 = a2 + c2 – 2acCosB c2 = a2 + b2 – 2abCosC

  21. Works for any Triangle S4 Credit Cosine Rule How to determine when to use the Cosine Rule. Two questions 1. Do you know ALL the lengths. SAS OR 2. Do you know 2 sides and the angle in between. www.mathsrevision.com If YES to any of the questions then Cosine Rule Otherwise use the Sine Rule Created by Mr Lafferty Maths Dept

  22. L 5m 43o 12m a2 = b2 + c2 -2bccosAo Using The Cosine Rule Works for any Triangle S4 Credit Example 1 : Find the unknown side in the triangle below: Identify sides a,b,c and angle Ao a = L b = 5 c = 12 Ao = 43o www.mathsrevision.com Write down the Cosine Rule. Substitute values to find a2. a2 = 52 + 122 - 2 x 5 x 12 cos 43o a2 = 25 + 144 - (120 x 0.731 ) a2 = 81.28 Square root to find “a”. a = L = 9.02m

  23. 17.5 m 137o 12.2 m M a2 = b2 + c2 -2bccosAo Using The Cosine Rule Works for any Triangle S4 Credit Example 2 : Find the length of side M. Identify the sides and angle. a = M b = 12.2 C = 17.5 Ao = 137o Write down Cosine Rule www.mathsrevision.com a2 = 12.22 + 17.52 – ( 2 x 12.2 x 17.5 x cos 137o ) a2 = 148.84 + 306.25 – ( 427 x – 0.731 ) Notice the two negative signs. a2 = 455.09 + 312.137 a2 = 767.227 a = M = 27.7m

  24. 43cm (1) 78o 31cm L (2) M 5.2m 38o 8m What Goes In The Box ? S4 Credit Find the length of the unknown side in the triangles: L = 47.5cm www.mathsrevision.com M =5.05m

  25. Cosine Rule S4 Credit Now try MIA Ex4.1 Ch12 (page 254) www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

  26. S4 Credit Starter Questions www.mathsrevision.com 54o Created by Mr. Lafferty Maths Dept.

  27. Cosine Rule S4 Credit Learning Intention Success Criteria • Know when to use the cosine rule to solve REAL LIFE problems. • 1. To show when to use the cosine rule to solve REAL LIFE problems involving finding an angle of a triangle . • 2. Solve REAL LIFE problems that involve finding an angle of a triangle. www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

  28. Works for any Triangle S4 Credit Cosine Rule The Cosine Rule can be used with ANY triangle as long as we have been given enough information. B a www.mathsrevision.com c C b A Created by Mr Lafferty Maths Dept

  29. a2 = b2 + c2 -2bccosAo Finding Angles Using The Cosine Rule Works for any Triangle S4 Credit Consider the Cosine Rule again: We are going to change the subject of the formula to cos Ao b2 + c2 – 2bc cos Ao = a2 Turn the formula around: Take b2 and c2 across. -2bc cos Ao = a2 – b2 – c2 www.mathsrevision.com 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.

  30. 9cm 11cm Ao 16cm Finding Angles Using The Cosine Rule Works for any Triangle S4 Credit Example 1 : Calculate the unknown angle Ao . Write down the formula for cos Ao a = 11 b = 9 c = 16 Label and identify Ao and a , b and c. Ao = ? www.mathsrevision.com Substitute values into the formula. Cos Ao = 0.75 Calculate cos Ao . Use cos-1 0.75 to find Ao Ao = 41.4o

  31. yo 13cm 15cm 26cm Finding Angles Using The Cosine Rule Works for any Triangle S4 Credit Example 2: Find the unknown Angle yo in the triangle: Write down the formula. a = 26 b = 15 c = 13 Ao = yo www.mathsrevision.com Identify the sides and angle. Find the value of cosAo The negative tells you the angle is obtuse. cosAo = - 0.723 Ao = yo = 136.3o

  32. (1) Ao 7m 5m 10m (2) 12.7cm 8.3cm 7.9cm What Goes In The Box ? S4 Credit Calculate the unknown angles in the triangles below: Bo www.mathsrevision.com Bo = 37.3o Ao =111.8o

  33. Cosine Rule S4 Credit Now try MIA Ex 5.1 Ch12 (page 256) www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

  34. S4 Credit Starter Questions www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

  35. S4 Credit Area of ANY Triangle Learning Intention Success Criteria • Know the formula for the area of any triangle. • 1. To explain how to use the Area formula for ANY triangle. • 2. Use formula to find area of any triangle given two length and angle in between. www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

  36. S4 Credit In Mathematics we have a convention for labelling triangles. Labelling Triangles B B a c C C www.mathsrevision.com b A A Small letters a, b, c refer to distances Capital letters A, B, C refer to angles Created by Mr Lafferty Maths Dept

  37. S4 Credit Have a go at labelling the following triangle. Labelling Triangles E E d f F F www.mathsrevision.com e D D Created by Mr Lafferty Maths Dept

  38. Co a b h Ao Bo c General Formula forArea of ANY Triangle S4 Credit Consider the triangle below: Area = ½ x base x height What does the sine of Ao equal www.mathsrevision.com Change the subject to h. h = b sinAo Substitute into the area formula

  39. Key feature To find the area you need to knowing 2 sides and the angle in between (SAS) S4 Credit The area of ANY triangle can be found by the following formula. Area of ANY Triangle B B a Another version c C C www.mathsrevision.com Another version b A A Created by Mr Lafferty Maths Dept

  40. S4 Credit Example : Find the area of the triangle. Area of ANY Triangle The version we use is B B 20cm c C C 30o www.mathsrevision.com 25cm A A Created by Mr Lafferty Maths Dept

  41. S4 Credit Example : Find the area of the triangle. Area of ANY Triangle The version we use is E 10cm 60o 8cm F www.mathsrevision.com D Created by Mr Lafferty Maths Dept

  42. (1) 12.6cm 23o 15cm (2) 5.7m 71o 6.2m Key feature Remember (SAS) What Goes In The Box ? S4 Credit Calculate the areas of the triangles below: A = 36.9cm2 www.mathsrevision.com A = 16.7m2

  43. Area of ANY Triangle S4 Credit Now try MIA Ex6.1 Ch12 (page 258) www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

  44. S4 Credit Starter Questions www.mathsrevision.com 61o Created by Mr. Lafferty Maths Dept.

  45. Mixed problems S4 Credit Learning Intention Success Criteria • Be able to recognise the correct trigonometric formula to use to solve a problem involving triangles. • 1. To use our knowledge gained so far to solve various trigonometry problems. www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

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

  47. L 57 miles 24 miles H 40 miles A B Exam Type Questions • 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. • Make a sketch of the journey. • Find the bearing of the lighthouse from the harbour. (nearest degree)

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

  49. P Not to Scale 670 miles 530 miles Q 520 miles W Exam Type Questions 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.

  50. Mixed Problems S4 Credit Now try MIA Ex 7.1 & 7.2 Ch12 (page 262) www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

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