IB DP Maths: AA SL复习笔记3.3.3 Applications of Trigonometry & Pythagoras
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
Exam 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.
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
Exam 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)Draw and label a diagram to show the top of the cliff, T, the foot of the cliff, F, the flag marker, M, and the boat, B, labelling all the angles and distances given above.
b)Find the distance from the boat to the foot of the cliff.
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
Exam 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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