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IB DP Maths: AA SL復習筆記3.3.3 Applications of Trigonometry & Pythagoras

Category: IB課程, 教材筆記, 福利干貨 Date: 2022年7月18日上午11:06

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

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.

3-3-3-ai-sl-bearings-we-solution-i

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.

3-3-3-ai-sl-elevation--depression-we-solution-i

b)Find the distance from the boat to the foot of the cliff.

3-3-3-ai-sl-elevation--depression-we-solution-ii

3-3-3-ai-sl-elevation--depression-we-solution-iii

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