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- 1. TrigonometryS4 Credit Sine Rule Finding a length www.mathsrevision.com Sine Rule Finding an Angle Cosine Rule Finding a Length Cosine Rule Finding an Angle Area of ANY Triangle Mixed Problems 23 Mar 2013 Created by Mr. Lafferty Maths Dept.
- 2. Starter QuestionsS4 Credit 1. Multiply out the brackets and simplify www.mathsrevision.com 5(y - 5) - 7(5 - y) 2. True or false the gradient of the line is 5 3 y = 5x - 4 3. Factorise x2 - 100 23 Mar 2013 Created by Mr. Lafferty Maths Dept.
- 3. Sine RuleS4 Credit Learning Intention Success Criteria www.mathsrevision.com 1. To show how to use the 1. Know how to use the sine sine rule to solve REAL rule to solve REAL LIFE LIFE problems involving problems involving lengths. finding the length of a side of a triangle . 23 Mar 2013 Created by Mr. Lafferty Maths Dept.
- 4. Sine RuleS4 Credit Works for any Triangle The Sine Rule can be used with ANY triangle as long as we have been given enough information. www.mathsrevision.com B a b c a = = SinA SinB SinC c C b A 23 Mar 2013 Created by Mr Lafferty Maths Dept
- 5. The Sine Rule Deriving the rule Consider a general triangle ABC. C CP SinB = ⇒ CP = aSinB a CP a b also SinA = ⇒ CP = bSinA b ⇒ aSinB = bSinA P aSinB B A ⇒ =b c SinA Draw CP perpendicular to BA a b ⇒ = SinA SinBThis can be extended to a b c SinA SinB SinC = = or equivalently = = SinA SinB SinC a b c
- 6. Calculating Sides Using The Sine RuleS4 Credit Example 1 : Find the length of a in this triangle. B www.mathsrevision.com 10m a 34o 41 o C A Match up corresponding sides and angles: a 10 a b c = = = sin 41o sin 34o sin Ao sin B sin C Rearrange and solve for a. 10sin 41o 10 × 0.656 a= a= = 11.74m sin 34o 0.559
- 7. Calculating SidesS4 Credit Using The Sine Rule Example 2 : Find the length of d in this triangle. D 10m www.mathsrevision.com 133 o 37o E C Match up corresponding sides and angles: d d 10 c d e o = = = sin133 sin 37 o sin C o sin D sin E Rearrange and solve for d. 10sin133o 10 × 0.731 d= d= = 12.14m sin 37o 0.602
- 8. What goes in the Box ?S4 Credit Find the unknown side in each of the triangles below: www.mathsrevision.com 12cm (1) (2) b 47o 32 o a 72o 93o 16mm A = 6.7cm B = 21.8mm 23 Mar 2013 Created by Mr Lafferty Maths Dept
- 9. Sine RuleS4 Credit www.mathsrevision.com Now try MIA Ex 2.1 Ch12 (page 247) 23 Mar 2013 Created by Mr. Lafferty Maths Dept.
- 10. Starter QuestionsS4 Credit 1. True or false 9x - 36 = 9(x + 6)(x - 6) www.mathsrevision.com 2. Find the gradient and the y - intercept 3 1 for the line with equation y = - x + 4 5 3. Solve the equation tanx - 1 = 0 23 Mar 2013 Created by Mr. Lafferty Maths Dept.
- 11. Sine RuleS4 Credit Learning Intention Success Criteria www.mathsrevision.com 1. To show how to use the 1. Know how to use the sine sine rule to solve problems rule to solve problems involving finding an angle involving angles. of a triangle . 23 Mar 2013 Created by Mr. Lafferty Maths Dept.
- 12. Calculating Angles Using The Sine RuleS4 Credit B Example 1 : 45m 38m Find the angle A o www.mathsrevision.com 23o C A Match up corresponding sides and angles: 45 38 a b c = = = sin Ao sin 23o sin A sin B sin C Rearrange and solve for sin Ao 45sin 23o sin A = o = 0.463 Use sin-1 0.463 to find Ao 38 Ao = sin −1 0.463 = 27.6o
- 13. Calculating AnglesS4 Credit Using The Sine Rule 75m Example 2 : X Z Find the angle Xo www.mathsrevision.com 143o 38m Y Match up corresponding sides and angles: x y z 38 75 = = = sin X sin Y sin Z sin X o sin143o Rearrange and solve for sin Xo 38sin143o sin X o = = 0.305 Use sin-1 0.305 to find Xo 75 −1 X = sin 0.305 = 17.8 o o
- 14. What Goes In The Box ?S4 Credit Calculate the unknown angle in the following: www.mathsrevision.com (1) 8.9m (2) 100o 12.9cm Bo Ao 14.5m 14o A = 37.2 o o 14.7cm Bo = 16o
- 15. Sine RuleS4 Credit www.mathsrevision.com Now try MIA Ex3.1 Ch12 (page 249) 23 Mar 2013 Created by Mr. Lafferty Maths Dept.
- 16. Starter QuestionsS4 Credit 1. Find the gradient of the line that passes www.mathsrevision.com through the points ( 1,1) and (9,9). 2. Find the gradient and the y - intercept for the line with equation y = 1 - x 3. Factorise x2 - 64 23 Mar 2013 Created by Mr. Lafferty Maths Dept.
- 17. Cosine RuleS4 Credit Learning Intention Success Criteria www.mathsrevision.com 1. To show when to use the 1. Know when to use the cosine cosine rule to solve rule to solve problems. problems involving finding the length of a side of a 2. Solve problems that involve triangle . finding the length of a side. 23 Mar 2013 Created by Mr. Lafferty Maths Dept.
- 18. Cosine RuleS4 Credit Works for any Triangle The Cosine Rule can be used with ANY triangle as long as we have been given enough information. www.mathsrevision.com a =b +c - 2bc cos A 2 2 2 B a c C b A 23 Mar 2013 Created by Mr Lafferty Maths Dept
- 19. The Cosine Rule The Cosine Rule generalises Pythagoras’ Theorem and 1 takes care of the 3 possible cases for Angle A. A Deriving the rule Consider a general triangle ABC. We require a in terms of b, c and A. B a2 = b2 + c2 BP = a – (b – x) 2 2 2 Also: BP2 = c2 – x2 2 c a ⇒ a2 – (b – x)2 = c2 – x2 ⇒ a2 – (b2 – 2bx + x2) = c2 – x2 A P ⇒ a2 – b2 + 2bx – x2 = c2 – x2A x b b-x C a2 > b2 + c2 b ⇒ a2 = b2 + c2 – 2bx* 3 ⇒ a = b + c – 2bcCosA 2 2 2Draw BP perpendicular to AC *Since Cos A = x/c ⇒ x = cCosA AWhen A = 90o, CosA = 0 and reduces to a2 = b2 + c2 1 Pythagoras When A > 90o, CosA is negative, ⇒ a2 > b2 + c2 Pythagoras + a bit 2 a2 < b2 + c2When A < 90o, CosA is positive, ⇒ a2 > b2 + c2 3 Pythagoras - a bit
- 20. The Cosine RuleThe 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 Finding an unknown side. a2 = b2 + c2 – 2bcCosA c aApplying the same method as A b C earlier to the other sidesproduce similar formulae for b2 = a2 + c2 – 2acCosB b and c. namely: c2 = a2 + b2 – 2abCosC
- 21. Cosine RuleS4 Credit Works for any Triangle How to determine when to use the Cosine Rule. www.mathsrevision.com Two questions 1. Do you know ALL the lengths. OR SAS 2. Do you know 2 sides and the angle in between. If YES to any of the questions then Cosine Rule Otherwise use the Sine Rule 23 Mar 2013 Created by Mr Lafferty Maths Dept
- 22. Using The Cosine RuleS4 Credit Works for any Triangle Example 1 : Find the unknown side in the triangle below: L www.mathsrevision.com 5m 43o 12m Identify sides a,b,c and angle Ao a= L b= 5 c = 12 Ao = 43o Write down the Cosine Rule. a2 = 52 + 122 - 2 x 5 x 12 cos 43o Substitute values to find a . 2 a2 = 25 + 144 - (120 x 0.731 ) a2 = 81.28 Square root to find “a”. a = L = 9.02m
- 23. Using The Cosine RuleS4 Credit Works for any Triangle 17.5 m Example 2 : 137 o 12.2 m Find the length of side M. www.mathsrevision.com M a = M b = 12.2 C = 17.5 Ao = 137o Identify the sides and angle. Write down Cosine Rule 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. What Goes In The Box ?S4 Credit Find the length of the unknown side in the triangles: 43cm www.mathsrevision.com (1) 78o 31cm L = 47.5cm L (2) 5.2m M 38o M =5.05m 8m
- 25. Cosine RuleS4 Credit www.mathsrevision.com Now try MIA Ex4.1 Ch12 (page 254) 23 Mar 2013 Created by Mr. Lafferty Maths Dept.
- 26. Starter QuestionsS4 Credit 1. If lines have the same gradient www.mathsrevision.com What is special about them. 2. Factorise x2 + 4x - 12 54o 3. Explain why the missing angles are 90 o and 36o 23 Mar 2013 Created by Mr. Lafferty Maths Dept.
- 27. Cosine RuleS4 Credit Learning Intention Success Criteria www.mathsrevision.com 1. To show when to use the 1. Know when to use the cosine cosine rule to solve REAL rule to solve REAL LIFE LIFE problems involving problems. finding an angle of a triangle . 2. Solve REAL LIFE problems that involve finding an angle of a triangle. 23 Mar 2013 Created by Mr. Lafferty Maths Dept.
- 28. Cosine RuleS4 Credit Works for any Triangle The Cosine Rule can be used with ANY triangle as long as we have been given enough information. www.mathsrevision.com a =b +c - 2bc cos A 2 2 2 B a c C b A 23 Mar 2013 Created by Mr Lafferty Maths Dept
- 29. Finding Angles Using The Cosine RuleS4 Credit Works for any Triangle Consider the Cosine Rule again: We are going to change the subject of the formula to cos A o www.mathsrevision.com b2 + c2 – 2bc cos Ao = a2 Turn the formula around: -2bc cos Ao = a2 – b2 – c2 Take b2 and c2 across. a2 − b2 − c2 Divide by – 2 bc. cos Ao = −2bc Divide top and bottom by -1 b +c −a 2 2 2 cos A = o You now have a formula for 2bc finding an angle if you know all three sides of the triangle.
- 30. Finding Angles Using The Cosine RuleS4 Credit Works for any Triangle Example 1 : Calculate the unknown angle Ao . www.mathsrevision.com b2 + c 2 − a 2 cos Ao = 2bc Write down the formula for cos Ao Ao = ? a = 11 b = 9 c = 16 Label and identify Ao and a , b and c. 92 + 16 2 − 112 cos A = o 2 × 9 × 16 Substitute values into the formula. Cos Ao = 0.75 Calculate cos Ao . Ao = 41.4o Use cos-1 0.75 to find Ao
- 31. Finding Angles Using The Cosine RuleS4 Credit Works for any Triangle Example 2: Find the unknown Angle yo in the triangle: www.mathsrevision.com b2 + c 2 − a 2 cos Ao = Write down the formula. 2bc A o = yo a = 26 b = 15 c = 13 Identify the sides and angle. 152 + 132 − 262 Find the value of cosAo cos Ao = 2 ×15 ×13 The negative tells you cosA = - 0.723 o the angle is obtuse. A o = yo = 136.3o
- 32. What Goes In The Box ?S4 Credit Calculate the unknown angles in the triangles below: www.mathsrevision.com (1) (2) 5m Ao 7m 12.7cm Bo 7.9cm 8.3cm 10m Ao =111.8o Bo = 37.3o
- 33. Cosine RuleS4 Credit www.mathsrevision.com Now try MIA Ex 5.1 Ch12 (page 256) 23 Mar 2013 Created by Mr. Lafferty Maths Dept.
- 34. Starter QuestionsS4 Credit 1. True or false www.mathsrevision.com 2( x + 3) − (4 − x) = 3 x − 2 2. Find the equaton of the line passing through the points ( 3,2) and (10, 9) . 3. Solve the equation sin x - 0.5 = 0 23 Mar 2013 Created by Mr. Lafferty Maths Dept.
- 35. Area of ANY TriangleS4 Credit Learning Intention Success Criteria www.mathsrevision.com 1. To explain how to use the 1. Know the formula for the Area formula for ANY area of any triangle. triangle. 2. Use formula to find area of any triangle given two length and angle in between. 23 Mar 2013 Created by Mr. Lafferty Maths Dept.
- 36. Labelling TrianglesS4 Credit In Mathematics we have a convention for labelling triangles. B www.mathsrevision.com a c C b A Small letters a, b, c refer to distances Capital letters A, B, C refer to angles 23 Mar 2013 Created by Mr Lafferty Maths Dept
- 37. Labelling TrianglesS4 Credit Have a go at labelling the following triangle. E www.mathsrevision.com d f F e D 23 Mar 2013 Created by Mr Lafferty Maths Dept
- 38. General Formula for Area of ANY TriangleS4 Credit Co Consider the triangle below: b a h www.mathsrevision.com Ao Bo c Area = ½ x base x height What does the sine of Ao equal 1 A = ×c×h h 2 sin A = o b 1 A = × c × b sin Ao Change the subject to h. 2 h = b sinAo 1 A = bc sin Ao Substitute into the area formula 2
- 39. Key feature Area of ANY TriangleS4 Credit To find the area you need to knowing The area sides and the angle be found 2 of ANY triangle can by the following formula. in between (SAS) www.mathsrevision.com B 1 Area = bc sin A a 2 Another version c C 1 Area = ac sin B 2 Another version b A 1 Area = ab sin C 23 Mar 2013 Created by Mr Lafferty Maths Dept 2
- 40. Area of ANY TriangleS4 Credit Example : Find the area of the triangle. www.mathsrevision.com The version we use is B 1 20cm Area = ab sin C 2 c 30o C 1 Area = × 20 × 25 × sin 30o 2 25cm A Area = 10 × 25 × 0.5 = 125cm 2 23 Mar 2013 Created by Mr Lafferty Maths Dept
- 41. Area of ANY TriangleS4 Credit Example : Find the area of the triangle. www.mathsrevision.com The version we use is E 1 10cm Area= df sin E 60o 2 8cm F 1 Area = × 8 ×10 × sin 60o 2 D Area = 40 × 0.866 = 34.64cm 2 23 Mar 2013 Created by Mr Lafferty Maths Dept
- 42. Key feature What Goes In The Box ? RememberS4 Credit (SAS) Calculate the areas of the triangles below: (1) www.mathsrevision.com 12.6cm A = 36.9cm2 23o 15cm (2) 5.7m A = 16.7m2 71 o 6.2m
- 43. Area of ANY TriangleS4 Credit www.mathsrevision.com Now try MIA Ex6.1 Ch12 (page 258) 23 Mar 2013 Created by Mr. Lafferty Maths Dept.
- 44. Starter QuestionsS4 Credit 1. A washing machine is reduced by 10% www.mathsrevision.com in a sale. Its sale price is £360. What was the original price. 2. Factorise x - 7x + 12 2 3. Find the missing angles. 61o 23 Mar 2013 Created by Mr. Lafferty Maths Dept.
- 45. Mixed problemsS4 Credit Learning Intention Success Criteria www.mathsrevision.com 1. To use our knowledge 1. Be able to recognise the gained so far to solve correct trigonometric various trigonometry formula to use to solve a problems. problem involving triangles. 23 Mar 2013 Created by Mr. Lafferty Maths Dept.
- 46. Exam Type Questions The angle of elevation of theAngle TDA = 180 – 35 = 145o top of a building measuredAngle DTA = 180 – 170 = 10o from point A is 25o. At point T D which is 15m closer to the building, the angle of elevation is 35o Calculate the height of the building. 10o 36.5 35o 145o 25o B D A 15 m t d a = =sin T sin D sin A SOH CAH TOA TD 15 TB = Sin 35o = Sin 25o Sin 10o 36.5 15Sin 25o ⇒ TB = 36.5Sin 35o = 20.9 m TD = = 36.5 m Sin 10
- 47. Exam Type QuestionsA 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) 572 + 402 − 24 2 CosA = 2x 57x 40 µ A = 20.4o L∴ Bearing = 90 − 20.4 = 070o 57 miles 24 miles H A 40 miles B
- 48. Exam Type Questions 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 TAngle BCA =180 – 110 = 70 Angle ACT = 180 – 70 = 110 Angle ATC = 180 – 115 = 65 o o o 65o t d a TC 53.21 110o = = = Csin T sin D sin A Sin 5o Sin 65o 70o 53.21 Sin 5 m⇒ TC = = 5.1 m (1dp ) 21 Sin 65o 53. 5o 25o SOH CAH TOA 20o A 50 m B 50 50 Cos 20o = ⇒ AC = = 53.21 m (2dp ) AC Cos 20o
- 49. Exam Type QuestionsAn AWACS aircraft takes off from RAF P Not to Scale 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. 670 miles Find the bearing of Q from point P. 530 miles b 2 + c 2 − a2 CosA = 2bc Q 5302 + 6702 − 5202 CosP = 2x 530x 670 520 miles µ W P = 48.7o ∴ Bearing = 180 + 48.7 = 229o
- 50. Mixed ProblemsS4 Credit www.mathsrevision.com Now try MIA Ex 7.1 & 7.2 Ch12 (page 262) 23 Mar 2013 Created by Mr. Lafferty Maths Dept.

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