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Trigonometry www.mathsrevision.com S5 Int2 Exact Values Angles greater than 90o Useful Notation & Area of a triangle Using Area of Triangle Formula Sine Rule Problems Cosine Rule Problems Mixed Problems 30-Apr-17 Created by Mr. Lafferty Maths Dept. Starter Questions www.mathsrevision.com S5 Int2 2 1. Factorise x - 36 2. A car depreciates at 20% each year. How much is it worth after 4 years if it cost £15 000 initially. 3. What sin30o as a fraction. 30-Apr-17 Created by Mr Lafferty Maths Dept Exact Values www.mathsrevision.com S5 Int2 Learning Intention 1. To build on basic trigonometry values. Success Criteria 1. Recognise basic triangles and exact values for sin, cos and tan 30o, 45o, 60o . 2. Calculate exact values for problems. 30-Apr-17 Created by Mr Lafferty Maths Dept Exact Values www.mathsrevision.com S5 Int2 Some special values of Sin, Cos and Tan are useful left as fractions, We call these exact values 60º 2 2 60º 60º 2 30º 2 3 1 This triangle will provide exact values for sin, cos and tan 30º and 60º 60º Exact Values www.mathsrevision.com S5 Int2 x Sin xº Cos xº Tan xº 0º 0 30º 45º 60º ½ 3 3 ½ 2 1 3 90º 2 3 Exact Values S5 Int2 www.mathsrevision.com Consider the square with sides 1 unit 1 1 45º 2 45º 1 1 We are now in a position to calculate exact values for sin, cos and tan of 45o Exact Values www.mathsrevision.com S5 Int2 x Sin xº Cos xº Tan xº 0º 0 30º 45º 60º ½ 1 2 3 3 2 1 2 ½ 1 3 1 3 90º 2 Exact Values www.mathsrevision.com S5 Int2 Now try Exercise 1 Ch8 (page 94) 30-Apr-17 Created by Mr Lafferty Maths Dept Starter Questions S5 Int2 www.mathsrevision.com 1. Write down the Compound Interest Formula and identify each term. 2. A house increases by 3% each year. How much is it worth in 5 years if it cost £40 000 initially. 3. What is the exact value of sin 45o. 30-Apr-17 Created by Mr Lafferty Maths Dept Angles Greater than 90o www.mathsrevision.com S5 Int2 Learning Intention 1. Introduce definition of sine, cosine and tangent over 360o using triangles with the unity circle. 30-Apr-17 Success Criteria 1. Find values of sine, cosine and tangent over the range 0o to 360o. 2. Recognise the symmetry and equal values for sine, cosine and tangent. Created by Mr. Lafferty Maths Dept. r S5 Int2 x y Angles Greater than 90o www.mathsrevision.com We will now use a new definition to cater for ALL angles. New Definitions y-axis y O 30-Apr-17 P(x,y) r Ao y sin A r x cos A x r tan A y x x-axis www.mathsrevision.com 11 Trigonometry www.mathsrevision.com S5 Int2 Angles over 900 Example 1 The radius line is 2cm. The point (1.2, 1.6). (1.2, 1.6) Find sin cos and tan for the angle. sin 53o 1.6 0.8 2 cos 53o 1.2 0.6 2 tan 53o 30-Apr-17 1.6 1.33 1.2 Check answer with calculator 53o Created by Mr Lafferty Maths Dept Trigonometry www.mathsrevision.com S5 Int2 Example 1 Angles over 900 The radius line is 2cm. The point (-1.8, 0.8). Find sin cos and tan for the angle. 0.8 sin127o 0.4 2 cos127o tan 53o 30-Apr-17 Check answer with calculator (-1.8, 0.8) 127o 1.8 0.9 2 0.8 0.44 1.8 Created by Mr Lafferty Maths Dept What Goes In The Box ? www.mathsrevision.com S5 Int2 Write down the equivalent values of the following in term of the first quadrant (between 0o and 90o): 1) Sin 135o sin 45o 1) Sin 300o - sin 60o 2) Cos 150o -cos 45o 2) Cos 360o cos 0o 3) Tan 135o -tan 45o 3) Tan 330o - tan 30o 4) Sin 225o -sin 45o 4) Sin 380o sin 20o 5) Cos 270o -cos 90o 5) Cos 460o - cos 80o Trigonometry Angles over 900 www.mathsrevision.com S5 Int2 Now try Exercise 2 Ch8 (page 97) 30-Apr-17 Created by Mr Lafferty Maths Dept Trigonometry Angles over 900 www.mathsrevision.com S5 Int2 Extension for unit 2 Trigonometry GSM Software 30-Apr-17 Created by Mr Lafferty Maths Dept Angles Greater than 90o www.mathsrevision.com Int 2 Two diagrams display same data in a different format Sin +ve 180o - xo (0,1) A0 (-1,0) 180o + xo Tan +ve created by Mr. Lafferty All +ve (1,0) 360o - xo (0,-1) Cos +ve Starter Questions S5 Int2 www.mathsrevision.com 1. Find the area of the triangle. 3cm 2. Factorise x - 4x + 3 2 8cm 3. Find the exact value of cos 120o. 30-Apr-17 Created by Mr. Lafferty Maths Dept. Area of a Triangle www.mathsrevision.com S5 Int2 Learning Intention Success Criteria 1. To show the standard way of labelling a triangle. 1. Be able to label a triangle properly. 2. Find the area of a triangle using basic trigonometry knowledge. 2. Find the area of a triangle using basic trigonometry knowledge. 30-Apr-17 Created by Mr. Lafferty Maths Dept. Labelling Triangles S5 Int2 www.mathsrevision.com In Mathematics we have a convention for labelling triangles. B a c A 30-Apr-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 S5 Int2 www.mathsrevision.com Have a go at labelling the following triangle. E d f D 30-Apr-17 F e Created by Mr Lafferty Maths Dept Area of a Triangle S5 Int2 Example 1 : Find the area of the triangle ABC. B 10cm A 50o 30-Apr-17 (i) Draw in a line from B to AC (ii) Calculate height BD BD Sin50 = 10 o 7.66cm D 12cm BD = 10 Sin50o = 7.66 (iii) C Area 1 Area base height 2 0.5 12 7.66 46cm 2 Created by Mr Lafferty Maths Dept Area of a Triangle S5 Int2 Example 2 : Find the area of the triangle PQR. P 12cm Q 40o 30-Apr-17 (i) Draw in a line from P to QR (ii) Calculate height PS PS Sin40 = 10 o 7.71cm S 20cm PS = 12 Sin40o = 7.71 (iii) R Area 1 Area base height 2 0.5 20 7.71 77.1cm2 Created by Mr Lafferty Maths Dept Constructing Pie Charts www.mathsrevision.com S5 Int2 Now try Exercise 3 Ch8 (page 99) 30-Apr-17 Created by Mr. Lafferty Maths Dept. Starter Questions www.mathsrevision.com S5 Int2 1. Multiply out and simplify 2( x 3) (4 x) 2. Find the volume of a cylinder 15cm in height and 10cm in diameter. 3. Write down the two values for sin that give a value of 0.5 30-Apr-17 Created by Mr. Lafferty Maths Dept. Area of ANY Triangle www.mathsrevision.com S5 Int2 Learning Intention 1. To explain how to use the Area formula for ANY triangle. 30-Apr-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. General Formula for Area of ANY Triangle Co 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 knowing The area of ANY can be found 2 sides andtriangle the angle byinthe following formula. between (SAS) www.mathsrevision.com S5 Int2 B a c A 30-Apr-17 1 Area= bc sin A 2 C Another version 1 Area= ac sin B 2 Another version b Created by Mr Lafferty Maths Dept 1 Area= ab sin C 2 Area of ANY Triangle S5 Int2 www.mathsrevision.com Example : Find the area of the triangle. B c A 30-Apr-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 S5 Int2 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 30-Apr-17 Created by Mr Lafferty Maths Dept Key feature What Goes In The Box ? Remember (SAS) S5 Int2 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 Area of ANY Triangle www.mathsrevision.com S5 Int2 Now try Exercise 4 Ch8 (page 100) 30-Apr-17 Created by Mr. Lafferty Maths Dept. Starter Questions S5 Int2 www.mathsrevision.com 1. Multiply out the brackets and simplify 5(y - 5) - 7(5 - y) 2. Find the gradient and the y - intercept 3 for the line with equation y = 5x 4 3. Factorise x2 - 100 30-Apr-17 Created by Mr. Lafferty Maths Dept. Sine Rule www.mathsrevision.com S5 Int2 Learning Intention 1. To show how to use the sine rule to solve REAL LIFE problems involving finding the length of a side of a triangle . 30-Apr-17 Success Criteria 1. Know how to use the sine rule to solve REAL LIFE problems involving lengths. Created by Mr. Lafferty Maths Dept. Sine Rule Works for any Triangle www.mathsrevision.com S5 Int2 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 30-Apr-17 a Created by Mr Lafferty Maths Dept C b 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 S4 Credit Example 1 : Find the length of a in this triangle. B a 10m A 34o 41o C Match up corresponding sides and angles: a b c sin Ao sin B sin C 10 a o sin 34 o sin 41 Rearrange and solve for a. 10sin 41o a sin 34o a 10 0.656 11.74m 0.559 www.mathsrevision.com S4 Credit 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 What goes in the Box ? S5 Int2 www.mathsrevision.com Find the unknown side in each of the triangles below: 12cm (1) a (2) 47o 32o 72o 93o a = 6.7cm 30-Apr-17 b Created by Mr Lafferty Maths Dept 16mm b = 21.8mm Sine Rule www.mathsrevision.com S5 Int2 Now try Ex 6&7 Ch8 (page 103) 30-Apr-17 Created by Mr. Lafferty Maths Dept. Starter Questions S5 Int2 www.mathsrevision.com 1. Factorise 9x - 36 2. Find the gradient and the y - intercept 1 3 for the line with equation y = - x + 5 4 3. Write down the two values of cos 1 that give you a value of 2 30-Apr-17 Created by Mr. Lafferty Maths Dept. Sine Rule www.mathsrevision.com S5 Int2 Learning Intention 1. To show how to use the sine rule to solve problems involving finding an angle of a triangle . 30-Apr-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 S4 Credit 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 C Calculating Angles Using The Sine Rule S4 Credit 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 What Goes In The Box ? S5 Int2 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 Sine Rule www.mathsrevision.com S5 Int2 Now try Ex 8 & 9 Ch8 (page 106) 30-Apr-17 Created by Mr. Lafferty Maths Dept. Starter Questions www.mathsrevision.com S5 Int2 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 30-Apr-17 Created by Mr. Lafferty Maths Dept. Cosine Rule www.mathsrevision.com S5 Int2 Learning Intention 1. To show when to use the cosine rule to solve problems involving finding the length of a side of a triangle . 30-Apr-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 S5 Int2 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 30-Apr-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. 2. An unknown angle when 3 sides are given. 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 S5 Int2 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 30-Apr-17 Created by Mr Lafferty Maths Dept Using The Cosine Rule Works for any Triangle S5 Int2 www.mathsrevision.com Example 1 : Find the unknown side in the triangle below: L 5m 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 S5 Int2 12.2 m www.mathsrevision.com Example 2 : Find the length of side M. a = M b = 12.2 C = 17.5 a2 = b2 + c2 -2bccosAo 137 17.5 m o M 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 What Goes In The Box ? S5 Int2 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 Cosine Rule www.mathsrevision.com S5 Int2 Now try Ex 12 Ch8 (page 112) 30-Apr-17 Created by Mr. Lafferty Maths Dept. Starter Questions S5 Int2 www.mathsrevision.com 1. If lines have the same gradient What is special about them. 2. Factorise x2 + 4x - 12 3. Find the missing angles. 30-Apr-17 Created by Mr. Lafferty Maths Dept. Cosine Rule www.mathsrevision.com S5 Int2 Learning Intention 1. To show when to use the cosine rule to solve REAL LIFE problems involving finding an angle of a triangle . 30-Apr-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 S5 Int2 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 30-Apr-17 Created by Mr Lafferty Maths Dept C b Finding Angles Using The Cosine Rule Works for any Triangle S5 Int2 www.mathsrevision.com Consider the Cosine Rule again: a2 = b2 + c2 -2bc cosAo 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 S5 Int2 www.mathsrevision.com Example 1 : Calculate the unknown angle xo . 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 Finding Angles Using The Cosine Rule Works for any Triangle S5 Int2 www.mathsrevision.com Example 2: Find the unknown yo 15cm 13cm Angle 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 = - 0.723 Ao = yo = 136.3o Identify the sides and angle. Find the value of cosAo The negative tells you the angle is obtuse. What Goes In The Box ? S5 Int2 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 Cosine Rule www.mathsrevision.com S5 Int2 Now try Ex 13 Ch8 (page 114) 30-Apr-17 Created by Mr. Lafferty Maths Dept. Starter Questions S5 Int2 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. 30-Apr-17 Created by Mr. Lafferty Maths Dept. 61o Mixed problems www.mathsrevision.com S5 Int2 Learning Intention 1. To use our knowledge gained so far to solve various trigonometry problems. 30-Apr-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 Mixed Problems www.mathsrevision.com S5 Int2 Now try Ex 14 Ch8 (page 117) 30-Apr-17 Created by Mr. Lafferty Maths Dept.