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Bearings (Cambridge O Level Maths)
Revision Note
Bearings
What are bearings?
- Bearings are a way of describing and using directions as angles
- They are specifically defined for use in navigation because they give a precise location and/or direction
How are bearings defined?
- There are three rules which must be followed every time a bearing is defined
- They are measured from the North direction
North is usually straight up in terms of a scale drawing or map drawn on a piece of paper and should be shown somewhere on the diagram
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They are measured clockwise (from North)
If you get muddled up look at a clock on the wall -
The angle should always be written (said) with 3 figures
So angles under 100° should have zero to fill in the missing figures, eg 059, 008
- They are measured from the North direction
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Notice also that the degree symbols are not usually included when talking about bearings
What are bearings used for?
- Bearings questions will normally involve the use of Pythagoras or trigonometry to find missing distances (lengths) and directions (angles) within navigation questions
- You should always draw a diagram
- There may be a scale given or you may need to consider using a scale
- Some questions may involve the use of angle facts to find the missing directions
- To answer a question involving drawing bearings the following steps may help:
- STEP 1: Draw a diagram adding in any points and distances you have been given
- STEP 2: Draw a North line (arrow pointing vertically up) at the point you wish to measure the bearing from
- If you are given the bearing from A to B draw the North line at A
- STEP 3: Measure the angle of the bearing given from the North line in the clockwise direction
- STEP 4: Draw a line and add the point B at the given distance
- You will likely then need to use Pythagoras's theorem or trigonometry to calculate another distance
How are bearings used on maps?
- If you need to use a map to find a direction, use a protractor to find the angle
- Always begin by drawing the North line
- Measure the angle clockwise from North
- Write the angle using 3 figures
- Knowing the compass directions for the common directions is helpful
- Due east means on a bearing of 090°
- Draw the line directly to the right
- Due south means on a bearing of 180°
- Draw the line vertically downwards
- Due west means on a bearing of 270°
- Draw the line directly to the left
- Due north means on a bearing of 360° (or 000°)
- Draw the line vertically upwards
- Due east means on a bearing of 090°
- Other directions can be found using these, for example
- Due Northeast means on a bearing of 045°
- This is halfway between North (000°) and East (090°)
- Due Southeast means on a bearing of 135°
- Due Southwest means on a bearing of 225°
- Due Northwest means on a bearing of 315°
- Due Northeast means on a bearing of 045°
- Using the above bearings for compass directions will help you to estimate angles for other bearings on the map
Examiner Tip
- Make sure you have all the equipment you need for your maths exams, along with a spare pen and pencil
- A rubber and pencil sharpener can be essential on these questions as they are all about accuracy
- Make sure you have compasses that aren’t loose and wobbly
- Make sure you can see and read the markings on your ruler and protractor
- Always draw a big, clear diagram and annotate it, be especially careful to label the angles in the correct places!
Worked example
A ship sets sail from the point P, as shown on the map below.
It sails on a bearing of 105 until it reaches the point Q, 70 km away. The ship then changes path and sails on a bearing of 065 for a further 35km, where its journey finishes.
Show on the map below the point Q and the final position of the ship.
Draw in a north line at the point P.
Measure an angle of 105° clockwise from the north line.
Make sure you are accurate, carefully make a small but visible mark on the map.
Draw a line from P through the mark you have made. Make this line longer than you expect to need it to be so that you can easily measure along it accurately.
Use the scale given on the map (1 cm = 10 km) to work out the number of cm that would represent 70 km.
70 km = 70 ÷ 10 = 7 cm
Accurately measure 7 cm from the point P along the line and make a clear mark on the line.
This is the point Q.
A bearing of 065 means 65° clockwise from the North.
First, draw a north line at the point Q, then carefully measure an angle of 65° clockwise from this line. Make a mark and then draw a line from Q through this mark.
Using the scale, find the distance in cm along the line you will need to measure.
35 km = 35 ÷ 10 = 3.5 cm
Accurately measure 3.5 cm from the point Q along this new line and make a clear mark on the line.
This is the final position of the ship.
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