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Applications of Trigonometry & Pythagoras (DP IB Maths: AI HL)
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
- An arrow showing the North line should be included on the diagram
- They are measured clockwise
- The angle is always written in 3 figures
- If the angle is less than 100° the first digit will be a zero
- They are measured from the North direction
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
- However normally in IB you will be using triangle calculations to find the distances
- Some questions may also 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 trigonometry to calculate the shortest distance or another given distance
Examiner Tip
- Always draw a big, clear diagram and annotate it, be especially careful to label the angles in the correct places!
Worked example
The point B is 7 km from A on a bearing of 105°. The distance from B to C is 5 km and the bearing from B to C is 230°. Find the distance from A to C.
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Elevation & Depression
What are the angles of elevation and depression?
- If a person looks at an object that is not on the same horizontal line as their eye-level they will be looking at either an angle of elevation or depression
- If a person looks up at an object their line of sight will be at an angle of elevation with the horizontal
- If a person looks down at an object their line of sight will be at an angle of depression with the horizontal
- Angles of elevation and depression are measured from the horizontal
- Right-angled trigonometry can be used to find an angle of elevation or depression or a missing distance
- Tan is often used in real-life scenarios with angles of elevation and depression
- For example if we know the distance we are standing from a tree and the angle of elevation of the top of the tree we can use Tan to find its height
- Or if we are looking at a boat at to sea and we know our height above sea level and the angle of depression we can find how far away the boat is
Examiner Tip
- It may be useful to draw more than one diagram if the triangles that you are interested in overlap one another
Worked example
A cliff is perpendicular to the sea and the top of the cliff stands 24 m above the level of the sea. The angle of depression from the cliff to a boat at sea is 35°. At a point m up the cliff is a flag marker and the angle of elevation from the boat to the flag marker is 18°.
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Constructing Diagrams
What diagrams will I need to construct?
- In IB you will be expected to construct diagrams based on information given
- The information will include compass directions, bearings, angles
- Look out for the plane the diagram should be drawn in
- It will either be horizontal (something occurring at sea or on the ground)
- Or it will be vertical (Including height)
- Work through the statements given in the instructions systematically
What do I need to know?
- Your diagrams will be sketches, they do not need to be accurate or to scale
- However the more accurate your diagram is the easier it is to work with
- Read the full set of instructions once before beginning to draw the diagram so you have a rough idea of where each object is
- Make sure you know your compass directions
- 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°
- Using the above bearings for compass directions will help you to estimate angles for other bearings on your diagram
Examiner Tip
- Draw your diagrams in pencil so that you can easily erase any errors
Worked example
A city at B is due east of a city at A and A is due north of a city at E. A city at C is due south of B.
The bearing from A to D is 155° and the bearing from D to C is 30°.
The distance AB = 50 km, the distances BC = CD = 30 km and the distances DE = AE = 40 km.
Draw and label a diagram to show the cities A, B, C, D and E and clearly mark the bearings and distances given.
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