further mathematics

Further Mathematics

Geometry & Trigonometry

Summary

Introduction

In this lesson we will consider how we can choose the right technique to use for a given problem.

This will include…

1.Things to do when starting a question

2.Choosing the right technique

3.Things to check before you finish

1.Starting a question

Read the question carefully. Draw a diagram and list any values that have

been given. Add any extra information that can be easily

worked out using geometry laws Eg: If you have two angles in a triangle find the

third (180° – other two angles).

Convert from bearings to angles Double check the question for more information

Eg: for similar figures, which one is the original

2. Choosing the right approach

To get started we will divide all of the possible questions into five groups.

1. Problems involving perimeters

2. Problems involving areas

3. Problems involving volumes

4. Problems involving similar figures

5. Problems involving lengths and angles of triangles

2.1 Problems involving perimeter

Find the total distance around the outside of the shape.

For questions involving circles useC = 2πr

2.2 Problems involving area

Simple shapes Choose from the formulas on p332

Composite shapes Divide the shape into simple shapes

Total Surface Area of a 3D shape For common shapes choose from the formulas on p338 For other shapes draw a net and add the areas of each face

(p339) For triangles where base and height are not known

For problems involving Area, 2 sides, 1 angle useArea = ½ ab sin C

For problems involving Area, 3 sides use Heron’s Formula (see page 422)

2.3 Problems involving volume

Prisms Use Vprism= Area of cross section height

Pyramids & Cones Use Vpyramid = 1/3 Area of base height

Spheres Use Vsphere = 4/3πr3

Composite shapes Divide the shape into prisms, pyramids & cones

and spheres. Find the volume of each and add them to get the total.

Examples

Find the perimeter of this shape.

Find the area.

Examples

Find the total surface area.

Find the volume.

Examples

Find the area.

Find the area.

2.4 Problems involving similar figures

Proving similarity Use AAA, SSS (or for similar triangles SAS)

Finding the scale factor Use k = length on copy ÷ length on original

Finding lengths using k Use the ratios of corresponding sides or Use the scale factor (above).

Problems involving areas and volumes Use lsf = k, asf = k2 and vsf = k3

2.5 Problems involving lengths and angles of triangles

Right angled triangles For problems involving 3 sides use Pythagoras theorem For problems involving 2 sides and 1 angle use

Trigonometric ratios (SOHCAHTOA) Triangles that do not have a right angle

For problems involving 2 sides, 2 angles use the Sine rule. To find an obtuse angle use

obtuse angle = 180° - acute angle For problems involving 3 sides, 1 angle use the Cosine rule.

To find an unknown side: To find an unknown angle:

bc

acbA

2cos

222

Abccba cos2222

Examples

What is the angle at B?

What is the angle s?

Examples What is the angle of elevation?

What is the length of the unknown side?

3. Before you finish

Don’t forget the last step in the calculation Did you need to take the square root? Did you need to use an inverse trig function (sin-1,

cos-1 or tan-1) Have you shown the correct units? Have you used the right number of decimal

places? If the answer was an angle…

Should it be converted to a bearing? Should it be in degrees and minutes?

Have you answered the question?