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Trigonometry Nat 5 www.mathsrevision.com Revision (SOH)(CAH)(TOA) Area of ANY Triangle Sine Rule Finding a length Sine Rule Finding an Angle Cosine Rule Finding a Length Cosine Rule Finding an Angle Mixed Problems Exam Type Questions 8-May-17 Created by Mr. Lafferty Maths Dept. Starter Questions Nat 5 www.mathsrevision.com 1. Calculate 104 x 100 putting your answer in standard form. 2. Is this triangle right angled ? If yes, find the size of angle x . o If no find the area of the triangle. 6 xo 10 www.mathsrevision.com 8 Angles & Triangles www.mathsrevision.com Nat 5 Learning Intention 1. We are revising SOHCAHTOA process. Success Criteria 1. Know the tree ratios for SOHCAHTOA. 2. Use SOHCAHTOA to finding an angle or length given a right-angled triangle. 8-May-17 Created by Mr. Lafferty Maths Dept. The Three Ratios Nat 5 adjacent www.mathsrevision.com opposite Sine Tangent Cosine hypotenuse adjacent Sine adjacent Cosine opposite Cosine Tangent Sine hypotenuse opposite www.mathsrevision.com Sine hypotenuse Trigonometry www.mathsrevision.com Nat 5 Sin x° = Opp Hyp Cos x° = Adj Hyp O A S HC H Tan x° = O T A Opp Adj Trigonometry Nat 5 www.mathsrevision.com Process 1. Write down SOH CAH TOA 2. 3. Identify what you want to find what you know Copy this! SOH CAH TOA (4 marks) SOH CAH TOA SOH CAH TOA SOH CAH TOA (4marks) Trigonometry www.mathsrevision.com Nat 5 Now try N5 TJ Ex8.1 Q3 onwards Ch8 (page 71) Starter Questions www.mathsrevision.com Nat 5 1. True or false 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 8-May-17 Created by Mr. Lafferty Maths Dept. Area of ANY Triangle www.mathsrevision.com Nat 5 Learning Intention 1. We are learning how to apply the Area formula for ANY triangle. 8-May-17 Success Criteria 1. Know the formula for the area of any triangle. 2. Use formula to find area of any triangle given two length and angle in between. Created by Mr. Lafferty Maths Dept. Labelling Triangles Nat 5 www.mathsrevision.com In Mathematics we have a convention for labelling triangles. B a c A 8-May-17 C b Small letters a, b, c refer to distances Capital letters A, B, C refer to angles Created by Mr Lafferty Maths Dept Labelling Triangles Nat 5 www.mathsrevision.com Have a go at labelling the following triangle. E d f D 8-May-17 F e Created by Mr Lafferty Maths Dept General Formula for Area of ANY Triangle Co Nat 5 www.mathsrevision.com Consider the triangle below: Ao Area = ½ x base x height 1 A ch 2 1 A c b sin Ao 2 1 A bc sin Ao 2 b a h Bo c What does the sine of Ao equal h o sin A b Change the subject to h. h = b sinAo Substitute into the area formula Key feature Area of ANY Triangle To find the area you need to know The area of ANY can be found 2 sides andtriangle the angle byinthe following formula. between (SAS) www.mathsrevision.com Nat 5 B 1 Area = bc sin A 2 a c A 8-May-17 C b Another version 1 Area = ac sin B 2 Another version Demo Created by Mr Lafferty Maths Dept 1 Area = ab sin C 2 Area of ANY Triangle Nat 5 www.mathsrevision.com Example : Find the area of the triangle. B c A 8-May-17 The version we use is 1 Area = ab sin C 2 20cm 30o 25cm C 1 Area 20 25 sin 30o 2 Area 10 25 0.5 125cm2 Created by Mr Lafferty Maths Dept Area of ANY Triangle Nat 5 www.mathsrevision.com Example : Find the area of the triangle. The version we use is E 60o 8cm 1 Area= df sin E 2 10cm F 1 Area 8 10 sin 60o 2 D Area 40 0.866 34.64cm2 8-May-17 Created by Mr Lafferty Maths Dept Key feature What Goes In The Box ? Remember (SAS) Nat 5 www.mathsrevision.com Calculate the areas of the triangles below: (1) 12.6cm A = 36.9cm2 23o 15cm (2) 5.7m 71o 6.2m A = 16.7m2 Trigonometry www.mathsrevision.com Nat 5 Now try N5 TJ Ex 8.2 Ch8 (page 73) Starter Questions Nat 5 www.mathsrevision.com 1. Multiply out the brackets and simplify 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 8-May-17 Created by Mr. Lafferty Maths Dept. Sine Rule www.mathsrevision.com Nat 5 Learning Intention 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 . 8-May-17 Success Criteria 1. Know how to use the sine rule to solve REAL LIFE problems involving lengths showing ALL appropriate working. Created by Mr. Lafferty Maths Dept. Sine Rule Works for any Triangle www.mathsrevision.com Nat 5 The Sine Rule can be used with ANY triangle as long as we have been given enough information. B a b c = = SinA SinB SinC c A 8-May-17 a Created by Mr Lafferty Maths Dept C b Demo The Sine Rule Consider a general triangle ABC. Deriving the rule C b a B P CP CP aSinB a CP also SinA CP bSinA b aSinB bSinA aSinB b SinA a b SinA SinB SinB c Draw CP perpendicular to BA A This can be extended to a b c SinA SinB SinC or equivalently SinA SinB SinC a b c Calculating Sides Using The Sine Rule www.mathsrevision.com Nat 5 Example 1 : Find the length of a in this triangle. B a 10m A 34o 41o C Match up corresponding sides and angles: a 10 o sin 41 sin 34 o a b c sin Ao sin B sin C Rearrange and solve for a. 10sin 41o a sin 34o a 10 0.656 11.74m 0.559 Demo www.mathsrevision.com Nat 5 Calculating Sides Using The Sine Rule Example 2 : Find the length of d in this triangle. D 10m o 133 37o C E Match up corresponding sides and angles: d d 10 o sin133 sin 37 o c d e sin C o sin D sin E Rearrange and solve for d. 10sin133o d sin 37o d 10 0.731 0.602 = 12.14m Demo What goes in the Box ? Nat 5 www.mathsrevision.com Find the unknown side in each of the triangles below: 12cm (1) a (2) 47o 32o b 72o 93o 16mm A = 6.7cm B = 21.8mm 8-May-17 Created by Mr Lafferty Maths Dept Trigonometry www.mathsrevision.com Nat 5 Now try N5 TJ Ex 8.3 Ch8 (page 76) Starter Questions www.mathsrevision.com Nat 5 1. True or false 9x - 36 = 9(x + 6)(x - 6) 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 8-May-17 Created by Mr. Lafferty Maths Dept. Sine Rule www.mathsrevision.com Nat 5 Learning Intention 1. We are learning how to use the sine rule to solve problems involving finding an angle of a triangle . 8-May-17 Success Criteria 1. Know how to use the sine rule to solve problems involving angles. Created by Mr. Lafferty Maths Dept. Calculating Angles Using The Sine Rule Nat 5 B www.mathsrevision.com Example 1 : Find the angle 45m 38m Ao 23o A Match up corresponding sides and angles: 45 38 sin Ao sin 23o a b c sin A sin B sin C Rearrange and solve for sin Ao 45sin 23o sin A = 0.463 38 o 1 Use sin-1 0.463 to find Ao A sin 0.463 27.6 o o Demo C Calculating Angles Using The Sine Rule Nat 5 www.mathsrevision.com Example 2 : 75m X Find the angle Xo 143o Y Match up corresponding sides and angles: 75 38 o sin143o sin X Z 38m x y z sin X sin Y sin Z Rearrange and solve for sin Xo o 38sin143 sin X o 75 1 Use sin-1 0.305 to find Xo = 0.305 X sin 0.305 17.8 o o Demo What Goes In The Box ? Nat 5 www.mathsrevision.com Calculate the unknown angle in the following: (1) 100o 8.9m Ao (2) 12.9cm Bo 14.5m Ao = 37.2o 14o 14.7cm Bo = 16o Trigonometry www.mathsrevision.com Nat 5 Now try N5 TJ Ex 8.4 Ch8 (page 79) Starter Questions www.mathsrevision.com Nat 5 1. Find the gradient of the line that passes 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 8-May-17 Created by Mr. Lafferty Maths Dept. Cosine Rule www.mathsrevision.com Nat 5 Learning Intention 1. We are learning when to use the cosine rule to solve problems involving finding the length of a side of a triangle . 8-May-17 Success Criteria 1. Know when to use the cosine rule to solve problems. 2. Solve problems that involve finding the length of a side. Created by Mr. Lafferty Maths Dept. Cosine Rule Works for any Triangle www.mathsrevision.com Nat 5 The Cosine Rule can be used with ANY triangle as long as we have been given enough information. a =b +c - 2bc cos A 2 2 2 B a c A 8-May-17 Created by Mr Lafferty Maths Dept C b The Cosine Rule The Cosine Rule generalises Pythagoras’ Theorem and takes care of the 3 possible cases for Angle A. Deriving the rule B Consider a general triangle ABC. We require a in terms of b, c and A. BP2 – (b – x)2 Also: BP2 = c2 – x2 a c = a2 1 A a2 = b2 + c2 2 a2 – (b – x)2 = c2 – x2 a2 – (b2 – 2bx + x2) = c2 – x2 A x P b b b-x Draw BP perpendicular to AC C a2 – b2 + 2bx – x2 = c2 – x2 a2 = b2 + c2 – 2bx* a2 = b2 + c2 – 2bcCosA *Since Cos A = x/c x = cCosA When A = 90o, CosA = 0 and reduces to a2 = b2 + c2 1 Pythagoras When A > 90o, CosA is negative, a2 > b2 + c2 2 Pythagoras + a bit When A < 90o, CosA is positive, a2 > b2 + c2 3 Pythagoras - a bit A a2 > b2 + c2 3 A a2 < b2 + c2 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 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: a c A b b2 = a2 + c2 – 2acCosB c2 = a2 + b2 – 2abCosC C Cosine Rule Works for any Triangle Nat 5 www.mathsrevision.com 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. If YES to any of the questions then Cosine Rule Otherwise use the Sine Rule 8-May-17 Created by Mr Lafferty Maths Dept Using The Cosine Rule Works for any Triangle Nat 5 www.mathsrevision.com Example 1 : Find the unknown side in the triangle below: L 5m Demo 43o Identify sides a,b,c and angle Ao 12m a= L b= 5 a2 = b2 + c = 12 Ao = 43o c2 -2bccosAo Write down the Cosine Rule. 2 a2 = 52 + 122 - 2 x 5 x 12 cos 43o Substitute values to find a . a2 = 25 + 144 - (120 x 0.731 ) a2 = 81.28 a = L = 9.02m Square root to find “a”. Using The Cosine Rule Works for any Triangle Nat 5 www.mathsrevision.com Example 2 : 12.2 m 137o 17.5 m Find the length of side M. a = M b = 12.2 C = 17.5 a2 = b2 + c2 -2bccosAo Ao = 137o M 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 Demo What Goes In The Box ? Nat 5 www.mathsrevision.com Find the length of the unknown side in the triangles: 43cm (1) 78o 31cm L L = 47.5cm (2) M 5.2m M =5.05m 38o 8m Trigonometry www.mathsrevision.com Nat 5 Now try N5 TJ Ex 8.5 Ch8 (page 81) Starter Questions Nat 5 www.mathsrevision.com 1. If lines have the same gradient What is special about them. 2. Factorise x + 4x - 12 2 3. Explain why the missing angles are 90o and 36o 8-May-17 Created by Mr. Lafferty Maths Dept. Cosine Rule www.mathsrevision.com Nat 5 Learning Intention 1. We are learning when to use the cosine rule to solve REAL LIFE problems involving finding an angle of a triangle . 8-May-17 Success Criteria 1. Know when to use the cosine rule to solve REAL LIFE problems. 2. Solve REAL LIFE problems that involve finding an angle of a triangle. Created by Mr. Lafferty Maths Dept. Cosine Rule Works for any Triangle www.mathsrevision.com Nat 5 The Cosine Rule can be used with ANY triangle as long as we have been given enough information. a =b +c - 2bc cos A 2 2 2 B a c A 8-May-17 Created by Mr Lafferty Maths Dept C b Finding Angles Using The Cosine Rule Works for any Triangle Nat 5 www.mathsrevision.com Consider the Cosine Rule again: a2 = b2 + c2 -2bccosAo We are going to change the subject of the formula to cos Ao b2 + c2 – 2bc cos Ao = a2 Turn the formula around: -2bc cos Ao = a2 – b2 – c2 Take b2 and c2 across. 2 2 2 a b c cos Ao 2bc b c a cos A 2bc 2 o 2 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. Finding Angles Using The Cosine Rule Works for any Triangle Nat 5 www.mathsrevision.com Example 1 : Calculate the unknown angle Ao . 2 2 2 b c a cos Ao 2bc a = 11 b = 9 Ao = ? 92 162 112 cos A 2 9 16 9cm 11cm Ao 16cm Write down the formula for cos Ao c = 16 Label and identify Ao and a , b and c. o Substitute values into the formula. Cos Ao = 0.75 Calculate cos Ao . Ao = 41.4o Use cos-1 0.75 to find Ao Demo Finding Angles Using The Cosine Rule Works for any Triangle Nat 5 www.mathsrevision.com Example 2: Find the unknown yo 15cm 13cm Angle yo in the triangle: 26cm 2 2 2 b c a cos Ao 2bc Ao = yo a = 26 b = 15 Write down the formula. c = 13 2 2 2 15 13 26 cos Ao 2 15 13 cosAo = Ao = yo = - 0.723 136.3o Identify the sides and angle. Find the value of cosAo The negative tells you the angle is obtuse. Demo What Goes In The Box ? Nat 5 www.mathsrevision.com Calculate the unknown angles in the triangles below: (1) 5m Ao 10m Ao =111.8o 7m (2) 12.7cm Bo 8.3cm 7.9cm Bo = 37.3o Trigonometry www.mathsrevision.com Nat 5 Now try N5 TJ Ex 8.6 Ch8 (page 84) Starter Questions Nat 5 www.mathsrevision.com 1. A washing machine is reduced by 10% in a sale. It's sale price is £360. What was the original price. 2. Factorise x - 7x +12 2 3. Find the missing angles. 8-May-17 Created by Mr. Lafferty Maths Dept. 61o Mixed problems www.mathsrevision.com Nat 5 Learning Intention 1. We are learning to use our knowledge gained so far to solve various trigonometry problems. 8-May-17 Success Criteria 1. Be able to recognise the correct trigonometric formula to use to solve a problem involving triangles. Created by Mr. Lafferty Maths Dept. Exam Type Questions Angle TDA = 180 – 35 = 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. 145o Angle DTA = 180 – 170 = 10o T 36.5 B t d a sin T sin D sin A TD 15 Sin 25o Sin10o 15Sin 25o TD 36.5 m Sin10 10o 35o 145o 25o D 15 m A SOH CAH TOA Sin 35o TB 36.5 TB 36.5Sin 35o 20.9 m 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. (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 A 20.4o L Bearing 90 20.4 070o 57 miles H 24 miles A 40 miles B 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 T o o o Angle BCA =180 – 110 = 70 Angle ACT = 180 – 70 = 110 Angle ATC = 180 – 115 = 65 t d a sin T sin D sin A TC 65o 110o TC 53.21 Sin 5o Sin 65o C 70o 53.21 Sin 5 5.1 m (1dp ) Sin 65o 5o A 20o 25o SOH CAH TOA B 50 m Cos 20o 50 AC AC 50 Cos 20o 53.21 m (2dp ) 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. b2 c 2 a2 CosA 2bc 5302 6702 5202 CosP 2x 530x 670 P 48.7o Bearing 180 48.7 229o Not to Scale P 670 miles 530 miles Q 520 miles W Trigonometry www.mathsrevision.com Nat 5 Now try N5 TJ Ex 8.7 & 8.8 Ch8 (page 85)