Introduction to Elevation, Depression, and Bearings

Welcome to one of the most practical parts of Trigonometry! Have you ever wondered how a pilot knows exactly where to land, or how a surveyor measures the height of a skyscraper without climbing it? They use angles of elevation, angles of depression, and bearings.

In this chapter, we take the trigonometry skills you've already learned (like \( \text{SOH CAH TOA} \)) and apply them to real-world navigation and heights. These topics are frequently tested in both Paper 1 and Paper 2 of your Edexcel International GCSE (Specification B) exams, so mastering them is a great way to boost your grade!

Note: Before starting, make sure you are comfortable with basic Sine, Cosine, and Tangent ratios. If you need a refresher, check out the "Sine, cosine and tangent up to 180 degrees" chapter!

1. Angles of Elevation and Depression

The most important thing to remember here is that these angles are always measured from a horizontal line.

What is an Angle of Elevation?

Imagine you are standing on the ground looking straight ahead (the horizontal). If you look up at the top of a tree, the angle your eyes move through is the angle of elevation.

What is an Angle of Depression?

Imagine you are standing on top of a cliff looking straight ahead (the horizontal). If you look down at a boat in the sea, the angle your eyes move through is the angle of depression.

Common Mistake: Many students accidentally measure the angle from the vertical (like a wall or a person's body). Always draw a horizontal line first!

The "Z-Angle" Trick

Did you know that the angle of elevation from Point A to Point B is exactly the same size as the angle of depression from Point B to Point A?
This is because the horizontal lines are parallel, creating alternate angles (often called Z-angles).
\( \text{Angle of Elevation} = \text{Angle of Depression} \)

Quick Review:

  • Elevation: Looking UP from the horizontal.
  • Depression: Looking DOWN from the horizontal.
  • Both are measured in degrees (\( ^\circ \)) and decimals of a degree.

2. Working with Bearings

Bearings are a way of describing directions using numbers. In the Pearson Edexcel Specification B exam, you must follow three golden rules for bearings:

  1. Start from North: Always draw a North line at your starting point.
  2. Measure Clockwise: Always turn your protractor (or calculate your angle) in a clockwise direction.
  3. Use Three Digits: Bearings are written with three figures. For example, \( 45^\circ \) is written as \( 045^\circ \).

Calculating Bearings

You often need to use your knowledge of parallel lines to solve bearing problems. Since all North lines are parallel to each other:

  • Interior Angles: The angles between two North lines facing each other add up to \( 180^\circ \).
  • Alternate Angles: Look for the "Z" shape between North lines to find equal angles.

Example: If the bearing of B from A is \( 070^\circ \), what is the bearing of A from B (the "back bearing")?
You can calculate this by: \( 70^\circ + 180^\circ = 250^\circ \).
Memory Aid: If the bearing is less than \( 180^\circ \), add \( 180^\circ \). If it is more than \( 180^\circ \), subtract \( 180^\circ \).

3. Solving Problems in 2D

Most exam questions will ask you to find a distance or an angle. Here is a step-by-step guide to tackling these:

Step 1: Draw a clear diagram. Don't worry if you aren't an artist! Use a ruler to draw straight lines and label your points (e.g., \( P \) for Plane, \( T \) for Tower).

Step 2: Mark the "Right Angle." Most elevation and depression problems create right-angled triangles between the object, the ground, and the horizontal.

Step 3: Identify your sides. Label the Opposite, Adjacent, and Hypotenuse based on the angle you are given.

Step 4: Choose your ratio.

  • Use \( \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \) if you are dealing with heights and horizontal distances.
  • Use \( \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \) or \( \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \) if you are dealing with diagonal distances (like the length of a ladder or a flight path).

Step 5: Calculate. Use your calculator. Make sure it is in DEG mode!

4. Moving into 3D

Sometimes, a problem might involve a 3D situation, such as a flagpole standing in the corner of a rectangular field. Don't panic! The trick is to extract a 2D triangle from the 3D shape.

Key Strategy:

  1. Find a triangle on the ground (usually using Pythagoras' Theorem: \( a^2 + b^2 = c^2 \)) to find a horizontal length.
  2. Use that horizontal length as the "Adjacent" side for a vertical triangle to find the angle of elevation.

Note: Per the syllabus, you will not be asked to calculate the angle between two planes or the angle between a line and a plane in this specific section, but you may need to find lengths and angles within 3D structures using Sine and Cosine rules.

Top Tips for Exam Success

  • Show your working: Even if you get the final answer wrong, you can get marks for drawing the correct triangle or choosing the correct trig ratio.
  • Read the question carefully: Does it ask for the bearing of A from B or B from A? The word "from" tells you where to draw your North line!
  • Check your units: Ensure all lengths are in the same units (e.g., all meters) before you start calculating.
  • Realistic answers: If you calculate the height of a tree to be \( 500 \text{ km} \), you've likely made a mistake. Re-check your decimal places!

Key Takeaways

1. Horizontal is Key: Elevation and Depression are always measured from the horizontal.
2. North, Clockwise, 3-Digits: The three rules for bearings.
3. SOH CAH TOA: Your best friend for turning angles into distances.
4. Draw It Out: A good diagram is half the battle won.