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S3 Credit Trigonometry www.mathsrevision.com The Tangent Ratio The Tangent using Angle The Tangent Ratio in Action The Tangent (The Adjacent side) The Tangent (Finding Angle) The Sine of an Angle The Sine Ration In Action The Sine ( Finding the Hypotenuse) The Cosine of an Angle Mixed Problems S3 Credit Starter Questions www.mathsrevision.com 1. An F1 car can complete a lap in 2 mins. A lap is 5 miles in length. Show that the average speed is 150mph 2. The resistance (R) in copper wire is directly proportion to its length (L) and inversely to the square of its radius (r). Write down an formula connecting R, L and r. www.mathsrevision.com www.mathsrevision.com Angles & Triangles Learning Intention 1. To identify the hypotenuse, opposite and adjacent sides in a right angled triangle. Success Criteria 1. Understand the terms hypotenuse, opposite and adjacent in right angled triangle. 2. Work out Tan Ratio. 29-Apr-17 Created by Mr. Lafferty Maths Dept. www.mathsrevision.com S3 Credit Trigonometry Let’s Investigate! www.mathsrevision.com Trigonometry means “triangle” and “measurement”. We will be using right-angled triangles. Opposite www.mathsrevision.com S3 Credit Trigonometry x° Adjacent Mathemagic! Trigonometry Opposite www.mathsrevision.com S3 Credit 30° Adjacent Opposite = 0.6 Adjacent Try another! Trigonometry Opposite www.mathsrevision.com S3 Credit 45° Adjacent Opposite = 1 Adjacent www.mathsrevision.com S3 Credit Trigonometry For an angle of 30°, Opposite = 0.6 Adjacent Opposite is called the tangent of an angle. Adjacent We write tan 30° = 0.6 www.mathsrevision.com S3 Credit Trigonometry The ancient Greeks discovered this and repeated this for all possible angles. Tan 25° 0.466 Tan 26° 0.488 Tan 27° 0.510 Tan 28° 0.532 Tan 30° =0.554 0.577 Tan 29° Tan 30° 0.577 Tan 31° 0.601 Tan 32° 0.625 Tan 33° 0.649 Tan 34° 0.675 Accurate to 3 decimal places! www.mathsrevision.com S3 Credit Trigonometry Now-a-days we can use calculators instead of tables to find the Tan of an angle. On your calculator press Followed by 30, and press Tan = Notice that your calculator is incredibly accurate!! Accurate to 9 decimal places! www.mathsrevision.com S3 Credit Trigonometry What’s the point of all this??? Don’t worry, you’re about to find out! Trigonometry www.mathsrevision.com S3 Credit How high is the tower? Opp 60° 12 m Trigonometry Opposite www.mathsrevision.com S3 Credit Copy this! 60° 12 m Adjacent www.mathsrevision.com S3 Credit Trigonometry Opp Tan x° = Adj Opp Tan 60° = 12 12 x Tan 60° = Opp Opp =12 x Tan 60° = 20.8m (1 d.p.) Copy this! Trigonometry www.mathsrevision.com S3 Credit So the tower’s 20.8 m high! 20.8m Don’t worry, you’ll be trying plenty of examples!! Opp Tan x° = Adj Opposite www.mathsrevision.com S3 Credit Trigonometry x° Adjacent Example Trigonometry S3 Credit www.mathsrevision.com Find the height h h 65° 8m Opp SOH CAH TOA Opp Tan x° = Adj Tan 65° = h 8 8 x Tan 65° = h h = 8 x Tan 65° = 17.2m (1 d.p.) www.mathsrevision.com S3 Credit Trigonometry Class Group Identifying the Tan Ratio Ex 3.1 & Ex4.1 MIA Page 203 www.mathsrevision.com S3 Credit Starter Questions 1. Find the area and perimeter 10cm of the circle. 2. Is the triangle right angled at P. Q Explain your answer. 6cm 3. Factorise 2x2 3x 2 www.mathsrevision.com 10cm P 7cm R www.mathsrevision.com Angles & Triangles Learning Intention 1. To use tan of the angle to solve problems. Success Criteria 1. Write down tan ratio. 2. Use tan of an angle to solve problems. 29-Apr-17 Created by Mr. Lafferty Maths Dept. www.mathsrevision.com S3 Credit Using Tan to calculate angles www.mathsrevision.com Example Trigonometry S3 Credit www.mathsrevision.com Calculate the tan xo ratio P SOH CAH TOA Opp 18m R x° 12m Q Opp Tan x° = Adj Tan x° = 18 12 Tan x° = 1.5 S3 Credit Calculate the size of angle xo Trigonometry www.mathsrevision.com Tan x° = 1.5 How do we find x°? We need to use Tan ⁻¹on the calculator. Tan ⁻¹is written above To get this press 2nd Tan ⁻¹ Tan Followed by Tan S3 Credit Trigonometry www.mathsrevision.com Tan x° = 1.5 Press 2nd Enter 1.5 Tan ⁻¹ Tan = x = Tan ⁻¹1.5 = 56.3° (1 d.p.) Trigonometry www.mathsrevision.com S3 Credit Process 1. Identify Hyp, Opp and Adj 2. Write down ratio Tan xo = Opp Adj 3. Calculate xo 2nd Tan ⁻¹ Tan www.mathsrevision.com S3 Credit Trigonometry Now try Exercise 4.2 MIA Page 205 Starter Questions S3 Credit www.mathsrevision.com 1. True or false 30 5 + 6 4 = 48 2. Write in scientific notation 0.0456 3. Identify the sides of the triangle. xo 4. The subway train takes 6 mins to travel between 2 stations 3 miles apart. Show that it's average speed is 30mph. www.mathsrevision.com www.mathsrevision.com Angles & Triangles Learning Intention 1. To use tan of the angle to solve REAL LIFE problems. Success Criteria 1. Write down tan ratio. 2. Use tan of an angle to solve REAL LIFE problems. 29-Apr-17 Created by Mr. Lafferty Maths Dept. Trigonometry www.mathsrevision.com S3 Credit Use the tan ratio to find the height h of the tree to 2 decimal places. tan 47o = opp h = adj 8 tan 47o = h 8 SOH CAH TOA rod h = 8 × tan 47o h = 8.58m 29-Apr-17 47o Compiled by Mr. Lafferty Maths Dept. 8m www.mathsrevision.com S3 Credit Trigonometry SOH CAH TOA Example 2 Q1. An aeroplane is preparing to land at Glasgow Airport. It is over Lennoxtown at present which is 15km from the airport. The angle of descent is 6o. 29-Apr-17 What is the height of the plane ? tan 6o = h 15 c h = 15 × tan 6o h = 1.58km Aeroplane 6o Airport Compiled by Mr. Lafferty Maths Dept. a = 15 Lennoxtown www.mathsrevision.com S3 Credit Trigonometry Now try Exercise 5.1 MIA Page 207 Starter Questions www.mathsrevision.com S3 Credit 1. Explain why 6 + 9 3 = 9 and not 5 2. Write in scientific notation 32.56 3. Identify the sides of the triangle. xo 4. The train takes 10 mins to travel between 2 stations 6miles apart. Find the average speed of the train. www.mathsrevision.com www.mathsrevision.com Angles & Triangles Learning Intention 1. To use tan of the angle to find adjacent length. Success Criteria 1. Write down tan ratio. 2. Use tan of an angle to solve find adjacent length. 29-Apr-17 Created by Mr. Lafferty Maths Dept. Trigonometry www.mathsrevision.com S3 Credit Use the tan ratio to calculate how far the ladder is away from the building. opp 12 tan 45 = = adj d o 12 d= tan 45o SOH CAH TOA ladder 45o d = 12m 29-Apr-17 Compiled by Mr. Lafferty Maths Dept. dm 12m www.mathsrevision.com S3 Credit Trigonometry Example 2 Q1. An aeroplane is preparing to land at Glasgow Airport. It is over Lennoxtown at present. It is at a height of 1.58 km above the ground. It ‘s angle of descent is 6o. How far is it from the airport to Lennoxtown? tan 6o = 1.58 d SOH CAH TOA 1.58 d= tan 6o d = 15 km 29-Apr-17 Aeroplane a = 1.58 km 6o Airport Compiled by Mr. Lafferty Maths Dept. Lennoxtown www.mathsrevision.com S3 Credit Trigonometry Now try Exercise 5.2 MIA Page 210 Starter Questions www.mathsrevision.com S3 Credit 1. 3 3 5 + 4 5 7 2. Explain why y= y = 6 when a = (-1) b = 2 (a - b)(b - a)(2a - b) p . v Find k when T = 6 P = 18 and v = 9. Q3. Given T = k www.mathsrevision.com www.mathsrevision.com Angles & Triangles Learning Intention 1. To show how to find an angle using tan ratio. Success Criteria 1. Write down tan ratio. 2. Use tan ratio to find an angle. 29-Apr-17 Created by Mr. Lafferty Maths Dept. Trigonometry www.mathsrevision.com S3 Credit Use the tan ratio to calculate the angle that the support wire makes with the ground. opp 11 tan x = = adj 4 o SOH CAH TOA 11 x = tan 4 o -1 11m x o = 70o 29-Apr-17 Compiled by Mr. Lafferty Maths Dept. xo 4m Trigonometry www.mathsrevision.com S3 Credit Use the tan ratio to find the angle of take-off. SOH CAH TOA opp 88 tan x = = adj 500 o tan x o = 0.176 o -1 o x = tan (0.176) = 10 29-Apr-17 Compiled by Mr. Lafferty Maths Dept. 88m xo 500 m www.mathsrevision.com S3 Credit Trigonometry Now try Exercise 6.1 MIA Page 211 www.mathsrevision.com S3 Credit Starter Questions 1. A train takes 12 minutes to travel between 2 stations. Show that the average speed is 60km/hr if the stations are 6miles apart. 2. Calculate A when w = (-3) y = 4 A= (w - y) + (y - w) www.mathsrevision.com www.mathsrevision.com Angles & Triangles Learning Intention 1. Definite the sine ratio and show how to find an angle using this ratio. Success Criteria 1. Write down sine ratio. 2. Use sine ratio to find an angle. 29-Apr-17 Created by Mr. Lafferty Maths Dept. The Sine Ratio Sin x° = Opposite www.mathsrevision.com S3 Credit Trigonometry x° Opp Hyp www.mathsrevision.com S3 Credit Find the height h Example Trigonometry h Opp Opp Sin x° = Hyp Sin 34° = 11cm 34° h 11 SOH CAH TOA 11 x Sin 34° = h h = 11 x Sin 34° = 6.2cm (1 d.p.) www.mathsrevision.com S3 Credit Using Sin to calculate angles www.mathsrevision.com Example Trigonometry S3 Credit www.mathsrevision.com Find the xo 6m Opp 9m Opp Sin x° = Hyp 6 Sin x° = 9 x° SOH CAH TOA Sin x° = 0.667 (3 d.p.) S3 Credit Trigonometry www.mathsrevision.com Sin x° =0.667 (3 d.p.) How do we find x°? We need to use Sin ⁻¹on the calculator. Sin ⁻¹is written above To get this press 2nd Sin ⁻¹ Sin Followed by Sin Trigonometry S3 Credit www.mathsrevision.com Sin x° = 0.667 (3 d.p.) Press 2nd Enter 0.667 Sin ⁻¹ Sin = x = Sin ⁻¹0.667 = 41.8° (1 d.p.) www.mathsrevision.com S3 Credit Trigonometry Now try Exercise 7.1 MIA Page 212 S3 Credit Starter Questions www.mathsrevision.com 1. Explain why we can simply pick out Q1 , Q2 and Q3 then find the 5 figure summary for the data 19, 15, 11, 22, 9, 12, 11 2. Show that the original price of a car is £9000 If it costs £8100 after a discount of 10% 3. A lorry is travelling at 40mph. It has travelled 60 miles. How long has it taken to travel 60 miles. www.mathsrevision.com www.mathsrevision.com Angles & Triangles Learning Intention 1. To show how to use the sine ratio to solve Success Criteria 1. Write down sine ratio. REAL-LIFE problems. 2. Use sine ratio to solve REAL-LIFE problems. 29-Apr-17 Created by Mr. Lafferty Maths Dept. Trigonometry www.mathsrevision.com S3 Credit The support rope is 11.7m long. The angle between the rope and ground is 70o. Use the sine ratio to calculate the height of the flag pole. sin 70o = opp h = hyp 11.7 SOH CAH TOA h = 11.7 sin70o 11.7m h = 11m 29-Apr-17 Compiled by Mr. Lafferty Maths Dept. 70o h Trigonometry www.mathsrevision.com S3 Credit Use the sine ratio to find the angle of the ramp. SOH CAH TOA opp 10 sin x = = hyp 20 o 10 sin x = 20 o 10 o x = sin = 30 20 o 29-Apr-17 -1 20 m xo Compiled by Mr. Lafferty Maths Dept. 10m www.mathsrevision.com S3 Credit Trigonometry Now try Exercise 7.2 MIA Page 214 S3 Credit Starter Questions www.mathsrevision.com 1. Fill in the ? marks. 2 x 2 9 x 7 = (2 x ?)( x ?) 2. Calculate A when A= 2 w 4 www.mathsrevision.com w = (-10) www.mathsrevision.com Angles & Triangles Learning Intention 1. To show how to calculate the hypotenuse using the sine ratio. Success Criteria 1. Write down sine ratio. 2. Use sine ratio to find the hypotenuse. 29-Apr-17 Created by Mr. Lafferty Maths Dept. Example www.mathsrevision.com S3 Credit Trigonometry SOH CAH TOA Opp Sin x° = Hyp Sin 72° = r= 5 r 5 sin 72o r = 5.3 km A road AB is right angled at B. The road BC is 5 km. Calculate the length of the new road AC. B 5km C 72° r A www.mathsrevision.com S3 Credit Trigonometry Now try Exercise 8.1 MIA Page 215 www.mathsrevision.com S3 Credit Starter Questions 1. Explain why we can simply pick out Q1 and Q3 then find the 5 figure summary for the data 9, 5, 11, 2, 9, 2 2. Find the original price of a football If it costs £20 after a discount of 80% 3. A lorry is travelling at 50mph. It has travelled 75 miles. Show that the time taken is 1hr 30 mins. www.mathsrevision.com www.mathsrevision.com Angles & Triangles Learning Intention 1. Definite the cosine ratio and show how to find an length or angle using this ratio. Success Criteria 1. Write down cosine ratio. 2. Use cosine ratio to find a length or angle. 29-Apr-17 Created by Mr. Lafferty Maths Dept. The Cosine Ratio www.mathsrevision.com S3 Credit Trigonometry Cos x° = x° Adjacent Adj Hyp Example Trigonometry Adj Cos x° = Hyp b Cos 40° = 35 b 40° Opp www.mathsrevision.com S3 Credit Find the adjacent length b 35mm SOH CAH TOA 35 x Cos 40° = b b = 35 x Cos 40°= 26.8mm (1 d.p.) www.mathsrevision.com S3 Credit Using Cos to calculate angles www.mathsrevision.com Example Trigonometry 34cm Adj Cos x° = Hyp Cos x° = 34 45 Opp www.mathsrevision.com S3 Credit Find the angle xo x° 45cm SOH CAH TOA Cos x° = 0.756 (3 d.p.) x = Cos ⁻¹0.756 =41° www.mathsrevision.com S3 Credit Trigonometry Now try Exercise 9.1 MIA Page 216 S3 Credit Starter Questions 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 The Three Ratios S3 Credit 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 S3 Credit Sin x° = Opp Hyp Cos x° = Adj Hyp O A S HC H Tan x° = O T A Opp Adj Trigonometry www.mathsrevision.com S3 Credit Process 1. Write down SOH CAH TOA 2. 3. Identify what you want to find what you know Copy this! www.mathsrevision.com S3 Credit Trigonometry Past Paper Type Questions SOH CAH TOA www.mathsrevision.com S3 Credit Trigonometry Past Paper Type Questions SOH CAH TOA (4 marks) www.mathsrevision.com S3 Credit Trigonometry Past Paper Type Questions SOH CAH TOA www.mathsrevision.com S3 Credit Trigonometry Past Paper Type Questions SOH CAH TOA 4 marks www.mathsrevision.com S3 Credit Trigonometry Past Paper Type Questions SOH CAH TOA www.mathsrevision.com S3 Credit Trigonometry Past Paper Type Questions SOH CAH TOA (4marks) www.mathsrevision.com S3 Credit Trigonometry Past Paper Type Questions SOH CAH TOA www.mathsrevision.com S3 Credit Trigonometry Past Paper Type Questions SOH CAH TOA (4marks) www.mathsrevision.com Trigonometry www.mathsrevision.com Trigonometry www.mathsrevision.com S3 Credit Trigonometry Now try Exercise 10.1 & 10.2 MIA Page 218