IB DP Maths: AI HL复习笔记3.3.3 Applications of Trigonometry & Pythagoras

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

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

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

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
• Using the above bearings for compass directions will help you to estimate angles for other bearings on your diagram