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National 4 REL 1.3a REL 1.3b Trigonometry www.mathsrevision.com Investigating the Tangent Ratio The Tangent (Finding lengths) The Tangent (Finding the Angle) The Sine Ratio The Cosine Trickier Questions Mixed Problems / Exam Type Questions Starter Questions www.mathsrevision.com National 4 REL 1.3a REL 1.3b 2 66 % of £270 3 Q1. Calculate Q2. Convert 60% to fraction and simplify. Q3. Convert Q4. What is the time difference 09:28 and 10:50 Q5. The answer to the question is 90. What is the question. 11-May-17 4 5 to a percentage. Created by Mr.Lafferty Maths Dept Angles & Triangles www.mathsrevision.com National 4 REL 1.3a REL 1.3b Learning Intention 1. We are learning how to identify the hypotenuse, opposite and adjacent sides in a right angled triangle. 11-May-17 Success Criteria 1. Understand the terms hypotenuse, opposite and adjacent in right angled triangle. 2. Work out Tan Ratio. Created by Mr. Lafferty Maths Dept. Trigonometry Trigonometry means “triangle” and “measurement”. We will be using right-angled triangles. Opposite www.mathsrevision.com National 4 REL 1.3a REL 1.3b x° Adjacent These One triangles is 37o have The 2other things wein will common investigate Triangle Vertical side A 3cm B 6cm Horizontal side 4cm 8cm Vertical Horizontal 3 6 0.75 0.75 4 8 C 4cm D 1.5cm 2cm 6cm 4.5 3 0.75 0.75 6 4 Right-Angled Triangle www.mathsrevision.com National 4 REL 1.3a REL 1.3b Trigonometry For the RAT with angle of 37o the answer obtained by dividing vertical side by horizontal side was ALWAYS 0.75 tan37o The number has a special name It is called the TANGENT of 37o www.mathsrevision.com www.mathsrevision.com National 4 REL 1.3a REL 1.3b Using a calculator to find tan ratio What Goes In The Box ? Find the tan ratio to 3 decimal places. tan 0o tan 45o = 0.577 = 0.000 = 1.000 tan 60o tan 20o tan 70o = 1.732 = 0.364 = 2.747 tan 30o www.mathsrevision.com Trigonometry Opp Tan x° = Adj Opposite www.mathsrevision.com National 4 REL 1.3a REL 1.3b x° Adjacent Trigonometry Opposite www.mathsrevision.com National 4 REL 1.3a REL 1.3b Copy this! 60° 12 m Adjacent www.mathsrevision.com National 4 REL 1.3a REL 1.3b 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! www.mathsrevision.com National 4 REL 1.3a REL 1.3b Trigonometry Now Try TJ 4+ Ex19.1 Ch19 (Page 151) Starter Questions National 4 REL 1.3a REL 1.3b 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. 4. A train takes 6 hrs to travel 360 miles apart. Calculate it's average speed. www.mathsrevision.com xo Angles & Triangles www.mathsrevision.com National 4 REL 1.3a REL 1.3b Learning Intention 1. We are learning how to find a length using the tan ratio. Success Criteria 1. Write down tan ratio. 2. Use tan of an angle to solve REAL LIFE problems. 11-May-17 Created by Mr. Lafferty Maths Dept. Trigonometry Opp Tan x° = Adj Opposite www.mathsrevision.com National 4 REL 1.3a REL 1.3b x° Adjacent Example Find the height h Trigonometry www.mathsrevision.com National 4 REL 1.3 a + 1.3b 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.) Trigonometry www.mathsrevision.com National 4 REL 1.3a REL 1.3b 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 11-May-17 47o Compiled by Mr. Lafferty Maths Dept. 8m www.mathsrevision.com National 4 REL 1.3a REL 1.3b 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. 11-May-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 National 4 REL 1.3a REL 1.3b Trigonometry Now Try TJ 4+ Ex19.2 Ch19 (Page 151) www.mathsrevision.com National 4 REL 1.3a REL 1.3b 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 P www.mathsrevision.com 10cm 7cm R Angles & Triangles www.mathsrevision.com National 4 REL 1.3a REL 1.3b Learning Intention 1. We are learning how to find an angle using the tan ratio. Success Criteria 1. Write down tan ratio. 2. Use tan of an angle to solve problems. 11-May-17 Created by Mr. Lafferty Maths Dept. www.mathsrevision.com National 4 REL 1.3a REL 1.3b Using Tan to calculate angles www.mathsrevision.com www.mathsrevision.com National 4 REL 1.3a REL 1.3b Using a calculator to find angle given tan ratio What Goes In The Box ? Find the angle given the tan ratio to 1 decimal places. tan-1 0.5 tan-1 0.8 = 45.0o = 26.6o = 38.7o tan-1 0.6 tan-1 2 tan-1 3 = 31.0o = 64.4o = 71.6o tan-1 1 www.mathsrevision.com Example Trigonometry www.mathsrevision.com National 4 REL 1.3a REL 1.3b P SOH CAH TOA Opp 18m R Calculate the tan xo ratio x° 12m Q Opp Tan x° = Adj Tan x° = 18 12 Tan x° = 1.5 www.mathsrevision.com National 4 REL 1.3a REL 1.3b Calculate the size of angle xo Trigonometry 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 National 4 REL 1.3a REL 1.3b 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 National 4 REL 1.3a REL 1.3b Process 1. Identify Hyp, Opp and Adj 2. Write down ratio Tan xo = Opp Adj 3. Calculate xo 2nd Tan ⁻¹ Tan Trigonometry www.mathsrevision.com National 4 REL 1.3a REL 1.3b 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 11-May-17 Compiled by Mr. Lafferty Maths Dept. xo 4m www.mathsrevision.com National 4 REL 1.3a REL 1.3b Trigonometry Now Try TJ 4+ Ex19.3 Ch19 (Page 153) Starter Questions www.mathsrevision.com National 4 REL 1.3a REL 1.3b 1. Simplify a 4a 13d 11a a 3d 2. The ratio of apples to pears is 2 : 5. If there are 28 pieces of fruit in total. How many apples are there. 3. 11-May-17 Find HCF of 8 and 18 Created by Mr.Lafferty Maths Dept Angles & Triangles www.mathsrevision.com National 4 REL 1.3a REL 1.3b Learning Intention 1. We are learning the sine ratio and how to find a length or angle using this ratio. Success Criteria 1. Write down sine ratio. 2. Use sine ratio to find an angle. 11-May-17 Created by Mr. Lafferty Maths Dept. Trigonometry The Sine Ratio Sin x° = Opposite www.mathsrevision.com National 4 REL 1.3a REL 1.3b x° Opp Hyp www.mathsrevision.com National 4 REL 1.3a REL 1.3b Using a calculator to find sin ratio What Goes In The Box ? Find the sin ratio to 3 decimal places. sin 0o sin 45o = 0.500 = 0.000 = 0.707 sin 60o sin 20o sin 90o = 0.866 = 0.342 = 1.000 sin 30o www.mathsrevision.com www.mathsrevision.com National 4 REL 1.3a REL 1.3b Using a calculator to find angle given sin ratio What Goes In The Box ? Find the angle given the sin ratio to 1 decimal places. sin-1 0.5 sin-1 0.8 = 90.0o = 30.0o = 53.1o sin-1 0.6 sin-1 0.1 sin-1 0.9 = 36.9o = 5.7o = 64.2o sin-1 1 www.mathsrevision.com Find the National 4 height h Example Trigonometry www.mathsrevision.com REL 1.3a REL 1.3b 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.) Trigonometry www.mathsrevision.com National 4 REL 1.3a REL 1.3b 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 11-May-17 Compiled by Mr. Lafferty Maths Dept. 70o h Example www.mathsrevision.com National 4 REL 1.3a REL 1.3b 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 Example Trigonometry National 4 REL 1.3a REL 1.3b 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.) www.mathsrevision.com National 4 REL 1.3a REL 1.3b Trigonometry 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 National 4 REL 1.3a REL 1.3b 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.) Trigonometry www.mathsrevision.com National 4 REL 1.3a REL 1.3b 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 11-May-17 -1 20 m xo Compiled by Mr. Lafferty Maths Dept. 10m www.mathsrevision.com National 4 REL 1.3a REL 1.3b Trigonometry Now Try TJ 4+ Ex19.4 Ch19 (Page 155) Starter Questions National 4 REL 1.3a REL 1.3b www.mathsrevision.com 1. Simplify 5 y 6 x 7 x 3 x 3 xy 2. The ratio of apples to pears is 4 : 5. If there are 72 pieces of fruit in total. How many apples are there. 3. 11-May-17 Find HCF of 15 and 21 Created by Mr.Lafferty Maths Dept Angles & Triangles www.mathsrevision.com National 4 REL 1.3a REL 1.3b Learning Intention 1. We are learning the cosine ratio and how to find a length or angle using this ratio. Success Criteria 1. Write down cosine ratio. 2. Use cosine ratio to find a length or angle. 11-May-17 Created by Mr. Lafferty Maths Dept. www.mathsrevision.com National 4 REL 1.3a REL 1.3b Using a calculator to find cos ratio What Goes In The Box ? Find the cos ratio to 3 decimal places. cos 0o cos 45o = 0.866 = 1.000 = 0.707 cos 60o cos 20o cos 90o = 0.500 = 0.940 = 0.000 cos 30o www.mathsrevision.com www.mathsrevision.com National 4 REL 1.3a REL 1.3b Using a calculator to find angle given cos ratio What Goes In The Box ? Find the angle given the cos ratio to 1 decimal places. cos-1 0 cos-1 0.8 = 0.0o = 90.0o = 36.9o cos-1 0.6 cos-1 0.1 cos-1 0.9 = 53.1o = 84.3o = 25.8o cos-1 1 www.mathsrevision.com The Cosine Ratio www.mathsrevision.com National 4 REL 1.3a REL 1.3b Trigonometry Cos x° = x° Adjacent Adj Hyp Find the adjacent National 4 length b REL 1.3a Example Trigonometry b Adj Cos x° = Hyp b Cos 40° = 35 40° Opp www.mathsrevision.com REL 1.3b 35mm SOH CAH TOA 35 x Cos 40° = b b = 35 x Cos 40°= 26.8mm (1 d.p.) Example Trigonometry 34cm Adj Cos x° = Hyp Cos x° = 34 45 Opp www.mathsrevision.com National 4 REL 1.3a REL 1.3b Find the angle xo x° 45cm SOH CAH TOA Cos x° = 0.756 (3 d.p.) x = Cos ⁻¹0.756 =41° www.mathsrevision.com National 4 REL 1.3a REL 1.3b Trigonometry Now Try TJ 4+ Ex19.5 Ch19 (Page 158) Starter Questions National 4 REL 1.3a REL 1.3b www.mathsrevision.com 1. Simplify 5 y 6 x 7 x 3 x 3 xy 2. The ratio of apples to pears is 4 : 5. If there are 72 pieces of fruit in total. How many apples are there. 3. 11-May-17 Find HCF of 15 and 21 Created by Mr.Lafferty Maths Dept Angles & Triangles www.mathsrevision.com National 4 REL 1.3a REL 1.3b Learning Intention 1. We are learning to solve trickier problems using trig ratios. Success Criteria 1. Identify correct ratio. 2. Use balancing method to solve. 11-May-17 Created by Mr. Lafferty Maths Dept. Find the length b National 4 Example Trigonometry www.mathsrevision.com REL 1.3a REL 1.3b b Opp Tan x° = Adj 12cm Balancing Method Tan 30° = 12 b 12 b= Tan 30o 30° SOH CAH TOA b = 20. 8 cm ( to 1 decimal place) Find the length y National 4 Example Trigonometry www.mathsrevision.com REL 1.3a REL 1.3b Sin x° = Balancing Method Opp Hyp Sin 20° = 10 y 10 y= Sin 20o 20° 10m y SOH CAH TOA y = 29.2 m ( to 1 decimal place) Find the length k National 4 Example Trigonometry www.mathsrevision.com REL 1.3a REL 1.3b 14m Adj Cos x° = Hyp Balancing Method Cos 25° = 14 k 14 k= Cos 25o 25° k SOH CAH TOA k = 15.4 m ( to 1 decimal place) www.mathsrevision.com National 4 REL 1.3a REL 1.3b Trigonometry Now Try TJ 4+ Ex19.6 Ch19 (Page 160) National 4 REL 1.3a REL 1.3b 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 www.mathsrevision.com National 4 REL 1.3a REL 1.3b adjacent opposite Sine Tangent Cosine hypotenuse adjacent Sine adjacent Cosine opposite Cosine Tangent Sine hypotenuse opposite www.mathsrevision.com Sine hypotenuse Trigonometry www.mathsrevision.com National 4 REL 1.3a REL 1.3b Sin x° = Opp Hyp Cos x° = Adj Hyp O A S HC H Tan x° = O T A Opp Adj Trigonometry www.mathsrevision.com National 4 REL 1.3a REL 1.3b Process 1. Write down SOH CAH TOA 2. 3. Identify what you want to find what you know Copy this! www.mathsrevision.com National 4 REL 1.3a REL 1.3b Trigonometry Past Paper Type Questions SOH CAH TOA www.mathsrevision.com National 4 REL 1.3a REL 1.3b Trigonometry Past Paper Type Questions SOH CAH TOA (4 marks) www.mathsrevision.com National 4 REL 1.3a REL 1.3b Trigonometry Past Paper Type Questions SOH CAH TOA www.mathsrevision.com National 4 REL 1.3a REL 1.3b Trigonometry Past Paper Type Questions SOH CAH TOA 4 marks www.mathsrevision.com National 4 REL 1.3a REL 1.3b Trigonometry Past Paper Type Questions SOH CAH TOA www.mathsrevision.com National 4 REL 1.3a REL 1.3b Trigonometry Past Paper Type Questions SOH CAH TOA www.mathsrevision.com National 4 REL 1.3a REL 1.3b Trigonometry Past Paper Type Questions SOH CAH TOA (4marks) Trigonometry www.mathsrevision.com National 4 REL 1.3a REL 1.3b 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 11-May-17 Compiled by Mr. Lafferty Maths Dept. dm 12m www.mathsrevision.com National 4 REL 1.3a REL 1.3b Trigonometry Now Try TJ 4+ Ex19.7 Ch19 (Page 162)