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

Starter Questions www.mathsrevision.com S3 Credit

Created by Mr. Lafferty Maths Dept. Angles & Triangles Learning Intention Success Criteria To identify the hypotenuse, opposite and adjacent sides in a right angled triangle. 1. Understand the terms hypotenuse, opposite and adjacent in right angled triangle. www.mathsrevision.com 2. Work out Tan Ratio. 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Let’s Investigate! Trigonometry www.mathsrevision.com S3 Credit

Trigonometry means “triangle” and “measurement”. We will be using right-angled triangles. Opposite hypotenuse x° Adjacent

Mathemagic! Opposite hypotenuse 30° Adjacent Opposite = 0.6 Adjacent

Try another! Opposite hypotenuse 45° Adjacent Opposite = 1 Adjacent

Opposite Adjacent = 0.6 For an angle of 30°, Opposite Adjacent is called the tangent of an angle. We write tan 30° = 0.6

The ancient Greeks discovered this and repeated this for Credit Tan 25° 0.466 Tan 26° 0.488 Tan 27° 0.510 Tan 28° 0.532 Tan 29° 0.554 Tan 30° 0.577 Tan 31° 0.601 Tan 32° 0.625 Tan 33° 0.649 Tan 34° 0.675 The ancient Greeks discovered this and repeated this for all possible angles. Tan 30° = 0.577 Accurate to 3 decimal places!

On your calculator press Now-a-days we can use calculators instead of tables to find the Tan of an angle. On your calculator press Tan Followed by 30, and press = Notice that your calculator is incredibly accurate!! Accurate to 9 decimal places!

What’s the point of all this??? Don’t worry, you’re about to find out!

Opp How high is the tower? 60° 12 m

Copy this! Opposite hypotenuse 60° 12 m Adjacent

Opp Tan x° = Adj Opp Tan 60° = 12 12 x Tan 60° = Opp Opp = Copy this! Opp Tan x° = Adj Opp Tan 60° = 12 12 x Tan 60° = Opp Opp = 12 x Tan 60° = 20.8m (1 d.p.)

20.8m So the tower’s 20.8 m high! Don’t worry, you’ll be trying plenty of examples!!

S3 Credit Opp Tan x° = Adj Opposite x° Adjacent

SOH CAH TOA Opp Tan x° = Adj h Tan 65° = 8 8 x Tan 65° = h h = Find the height h Example S3 Credit SOH CAH TOA Opp Hyp Opp h Tan x° = Adj 65° h Tan 65° = 8m 8 Adj 8 x Tan 65° = h h = 8 x Tan 65° = 17.2m (1 d.p.)

Class Group Identifying the Tan Ratio Ex 3.1 & Ex4.1 MIA Page 203

Starter Questions www.mathsrevision.com 10cm Q 6cm 10cm P 7cm R S3 Credit 10cm www.mathsrevision.com Q 6cm 10cm P 7cm R www.mathsrevision.com

Created by Mr. Lafferty Maths Dept. Angles & Triangles Learning Intention Success Criteria To use tan of the angle to solve problems. 1. Write down tan ratio. www.mathsrevision.com 2. Use tan of an angle to solve problems. 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Using Tan to calculate angles Credit Using Tan to calculate angles www.mathsrevision.com www.mathsrevision.com

Calculate the tan xo ratio Example S3 Credit P SOH CAH TOA Opp Hyp Opp 18m Tan x° = Adj x° Q R 18 12m Tan x° = Adj 12 Tan x° = 1.5

We need to use Tan ⁻¹on the calculator. = 1.5 Tan x° Calculate the size of angle xo We need to use Tan ⁻¹on the calculator. = 1.5 Tan x° How do we find x°? Tan Tan ⁻¹ Tan ⁻¹is written above To get this press 2nd Followed by Tan

= 1.5 Tan x° Press Enter 1.5 = x = Tan ⁻¹1.5 = 56.3° (1 d.p.) Tan ⁻¹

Process 1. Identify Hyp, Opp and Adj 2. Write down ratio Tan xo = Opp 3. Calculate xo 2nd

Now try Exercise 4.2 MIA Page 205

Starter Questions www.mathsrevision.com xo S3 Credit

Created by Mr. Lafferty Maths Dept. Angles & Triangles Learning Intention Success Criteria To use tan of the angle to solve REAL LIFE problems. 1. Write down tan ratio. www.mathsrevision.com 2. Use tan of an angle to solve REAL LIFE problems. 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Compiled by Mr. Lafferty Maths Dept. Use the tan ratio to find the height h of the tree to 2 decimal places. SOH CAH TOA 47o 8m rod 19-Apr-17 Compiled by Mr. Lafferty Maths Dept.

Compiled by Mr. Lafferty Maths Dept. 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. What is the height of the plane ? Aeroplane c 6o Airport a = 15 Lennoxtown 19-Apr-17 Compiled by Mr. Lafferty Maths Dept.

Now try Exercise 5.1 MIA Page 207

Starter Questions www.mathsrevision.com xo S3 Credit

Created by Mr. Lafferty Maths Dept. Angles & Triangles Learning Intention Success Criteria To use tan of the angle to find adjacent length. 1. Write down tan ratio. www.mathsrevision.com 2. Use tan of an angle to solve find adjacent length. 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Compiled by Mr. Lafferty Maths Dept. Use the tan ratio to calculate how far the ladder is away from the building. SOH CAH TOA 45o 12m ladder 19-Apr-17 Compiled by Mr. Lafferty Maths Dept. d m

Compiled by Mr. Lafferty Maths Dept. 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? SOH CAH TOA Aeroplane a = 1.58 km 6o Airport Lennoxtown 19-Apr-17 Compiled by Mr. Lafferty Maths Dept.

Now try Exercise 5.2 MIA Page 210

Starter Questions www.mathsrevision.com S3 Credit

Created by Mr. Lafferty Maths Dept. Angles & Triangles Learning Intention Success Criteria To show how to find an angle using tan ratio. 1. Write down tan ratio. www.mathsrevision.com 2. Use tan ratio to find an angle. 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Compiled by Mr. Lafferty Maths Dept. Use the tan ratio to calculate the angle that the support wire makes with the ground. SOH CAH TOA xo 11m 19-Apr-17 Compiled by Mr. Lafferty Maths Dept. 4 m

SOH CAH TOA Use the tan ratio to find the angle of take-off. 88m xo 19-Apr-17 Compiled by Mr. Lafferty Maths Dept. 500 m

Now try Exercise 6.1 MIA Page 211

Starter Questions www.mathsrevision.com S3 Credit

Created by Mr. Lafferty Maths Dept. Angles & Triangles Learning Intention Success Criteria Definite the sine ratio and show how to find an angle using this ratio. 1. Write down sine ratio. www.mathsrevision.com 2. Use sine ratio to find an angle. 19-Apr-17 Created by Mr. Lafferty Maths Dept.

S3 Credit The Sine Ratio Opp Sin x° = Hyp Opposite hypotenuse x°

SOH CAH TOA Opp Sin x° = Hyp h Sin 34° = 11 = h 11 x Sin 34° h = Find the height h Example S3 Credit Hyp 11cm h Opp Opp Sin x° = 34° Hyp h Sin 34° = SOH CAH TOA 11 = h 11 x Sin 34° h = 11 x Sin 34° = 6.2cm (1 d.p.)

Using Sin to calculate angles Credit Using Sin to calculate angles www.mathsrevision.com www.mathsrevision.com

SOH CAH TOA Opp Sin x° = Hyp 6 Sin x° = 9 Sin x° = 0.667 (3 d.p.) Find the xo Example S3 Credit Hyp 9m 6m Opp x° Opp Sin x° = Hyp SOH CAH TOA 6 Sin x° = 9 Sin x° = 0.667 (3 d.p.)

We need to use Sin ⁻¹on the calculator. How do we find x°? =0.667 (3 d.p.) Sin x° We need to use Sin ⁻¹on the calculator. How do we find x°? Sin Sin ⁻¹ Sin ⁻¹is written above To get this press 2nd Followed by Sin

= 0.667 (3 d.p.) Sin x° Press Enter 0.667 = x = Sin ⁻¹0.667 2nd Enter 0.667 = x = Sin ⁻¹0.667 = 41.8° (1 d.p.)

Now try Exercise 7.1 MIA Page 212

Starter Questions www.mathsrevision.com S3 Credit

Created by Mr. Lafferty Maths Dept. Angles & Triangles Learning Intention Success Criteria To show how to use the sine ratio to solve REAL-LIFE problems. 1. Write down sine ratio. www.mathsrevision.com Use sine ratio to solve REAL-LIFE problems. 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Compiled by Mr. Lafferty Maths Dept. 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. SOH CAH TOA 70o h 11.7m 19-Apr-17 Compiled by Mr. Lafferty Maths Dept.

SOH CAH TOA Use the sine ratio to find the angle of the ramp. 20 m 10m xo 10m 20 m 19-Apr-17 Compiled by Mr. Lafferty Maths Dept.

Now try Exercise 7.2 MIA Page 214

Starter Questions www.mathsrevision.com S3 Credit

Created by Mr. Lafferty Maths Dept. Angles & Triangles Learning Intention Success Criteria To show how to calculate the hypotenuse using the sine ratio. 1. Write down sine ratio. www.mathsrevision.com 2. Use sine ratio to find the hypotenuse. 19-Apr-17 Created by Mr. Lafferty Maths Dept.

SOH CAH TOA r r r = r = Opp Sin x° = Hyp 5 Sin 72° = 5.3 km Example B Credit SOH CAH TOA A road AB is right angled at B. The road BC is 5 km. Calculate the length of the new road AC. Opp Sin x° = Hyp B A 5 72° Sin 72° = r 5km r r = r = C 5.3 km

Now try Exercise 8.1 MIA Page 215

Starter Questions www.mathsrevision.com S3 Credit

Created by Mr. Lafferty Maths Dept. Angles & Triangles Learning Intention Success Criteria Definite the cosine ratio and show how to find an length or angle using this ratio. 1. Write down cosine ratio. www.mathsrevision.com 2. Use cosine ratio to find a length or angle. 19-Apr-17 Created by Mr. Lafferty Maths Dept.

The Cosine Ratio S3 Credit Adj Cos x° = Hyp hypotenuse x° Adjacent

Find the adjacent length b Example S3 Credit b Adj 40° Adj Cos x° = Opp Hyp Hyp 35mm b Cos 40° = 35 SOH CAH TOA 35 x Cos 40° = b b = 35 x Cos 40° = 26.8mm (1 d.p.)

Using Cos to calculate angles Credit Using Cos to calculate angles www.mathsrevision.com www.mathsrevision.com

SOH CAH TOA Adj Cos x° = Hyp 34 Cos x° = 45 Cos x° = 0.756 (3 d.p.) Find the angle xo Example S3 Credit Adj 34cm x° Adj Cos x° = Opp Hyp Hyp 45cm 34 Cos x° = 45 SOH CAH TOA Cos x° = 0.756 (3 d.p.) x = Cos ⁻¹0.756 =41°

Now try Exercise 9.1 MIA Page 216

Starter Questions www.mathsrevision.com xo 10 6 8 S3 Credit

The Three Ratios www.mathsrevision.com adjacent opposite Tangent Credit adjacent opposite Tangent Cosine Sine hypotenuse Sine adjacent adjacent Cosine www.mathsrevision.com Cosine Tangent hypotenuse opposite opposite Sine Sine hypotenuse www.mathsrevision.com

SOH CAH TOA Sin x° = Opp Hyp Cos x° = Adj Hyp Tan x° = Opp Adj S3 Credit Sin x° = Opp Hyp Cos x° = Adj Hyp Tan x° = Opp Adj SOH CAH TOA

Process SOH CAH TOA 1. Write down 2. 3. what you know Credit Process 1. Write down SOH CAH TOA 2. Identify what you want to find what you know 3. Copy this!

Past Paper Type Questions Credit Past Paper Type Questions SOH CAH TOA

Past Paper Type Questions Credit Past Paper Type Questions SOH CAH TOA (4 marks)

Past Paper Type Questions Credit Past Paper Type Questions SOH CAH TOA

Past Paper Type Questions Credit Past Paper Type Questions SOH CAH TOA 4 marks

Past Paper Type Questions Credit Past Paper Type Questions SOH CAH TOA

Past Paper Type Questions Credit Past Paper Type Questions SOH CAH TOA (4marks)

Past Paper Type Questions Credit Past Paper Type Questions SOH CAH TOA

Past Paper Type Questions Credit Past Paper Type Questions SOH CAH TOA (4marks)

Now try Exercise 10.1 & 10.2 MIA Page 218