What you'll learn
This topic tests your ability to apply trigonometric ratios to solve real-world problems involving heights, distances, and navigation. CXC CSEC Mathematics papers consistently include questions on angles of elevation and depression, three-figure bearings, and combined problems requiring multiple trigonometric steps. Understanding these applications is essential for securing marks in Paper 2 Section II.
Key terms and definitions
Angle of elevation — the angle measured upward from the horizontal line of sight to an object above the observer's eye level.
Angle of depression — the angle measured downward from the horizontal line of sight to an object below the observer's eye level.
Bearing — the direction of one point from another, measured as an angle in degrees clockwise from north, always expressed as three figures (e.g., 045°, 273°).
True bearing — another term for three-figure bearing, distinguishing it from compass bearings (N45°E format).
Horizontal line of sight — the imaginary horizontal line drawn from the observer's eye, perpendicular to the vertical.
Back bearing — the bearing from point B to point A when the bearing from A to B is known; calculated by adding or subtracting 180° from the original bearing.
Trigonometric ratios — sine, cosine, and tangent relationships between angles and sides in right-angled triangles (SOH-CAH-TOA).
Clinometer — an instrument used to measure angles of elevation or depression, commonly used in surveying and navigation throughout the Caribbean region.
Core concepts
Understanding angles of elevation and depression
Angles of elevation and depression always form alternate angles with respect to parallel horizontal lines. When you stand on the ground looking up at the top of a building, your angle of elevation equals the angle of depression from the top of the building to you, because the horizontal line through your eyes is parallel to the horizontal line at the top of the building.
Key principles:
- Both angles are measured from the horizontal, never from the vertical
- The angle of elevation from point A to point B equals the angle of depression from point B to point A
- Always sketch a diagram showing the horizontal line clearly
- Label the right angle formed between the vertical and horizontal components
When solving problems:
- Draw a clear diagram with horizontal lines marked
- Identify which angle (elevation or depression) is given
- Mark the right-angled triangle formed
- Choose the appropriate trigonometric ratio based on the sides involved
- Solve for the unknown using algebraic manipulation
Setting up right-angled triangles from word problems
CXC CSEC Mathematics examiners expect you to extract information from worded scenarios and construct accurate diagrams. The ability to translate words into geometric representations distinguishes stronger candidates.
Steps for diagram construction:
- Identify the observer's position and mark it clearly
- Draw a horizontal line from the observer's eye level
- Mark the object being observed (tower, ship, bird)
- Connect the observer to the object with a line of sight
- Label the angle given (elevation or depression)
- Mark all known lengths and the unknown with appropriate variables
Common scenarios in CSEC papers:
- A person standing on a cliff observing a boat at sea
- Someone on the ground observing the top of a lighthouse or building
- An aircraft observing a ship or runway below
- A surveyor measuring the height of a coconut palm or radio tower
Applying trigonometric ratios (SOH-CAH-TOA)
Once you have established your right-angled triangle, select the correct ratio:
SOH: sin θ = Opposite/Hypotenuse Use when you know or need to find the side opposite the angle and the hypotenuse.
CAH: cos θ = Adjacent/Hypotenuse Use when you know or need to find the side adjacent to the angle and the hypotenuse.
TOA: tan θ = Opposite/Adjacent Use when you know or need to find the opposite and adjacent sides (most common in elevation/depression problems).
For finding angles when sides are known:
- Use sin⁻¹, cos⁻¹, or tan⁻¹ (inverse functions)
- Ensure your calculator is in degree mode
- Round angles to the nearest degree or one decimal place as specified
Three-figure bearings
Three-figure bearings measure direction clockwise from north, always using three digits. This system is standard in Caribbean maritime navigation, aviation, and surveying.
Rules for three-figure bearings:
- North = 000° (or 360°)
- East = 090°
- South = 180°
- West = 270°
- All bearings between 000° and 360°
- Always write with three figures: 005°, 045°, 090°, 135°, 270°
Calculating back bearings:
- If bearing from A to B is less than 180°: add 180°
- If bearing from A to B is greater than 180°: subtract 180°
- Example: bearing A to B = 065°, so B to A = 065° + 180° = 245°
- Example: bearing A to B = 215°, so B to A = 215° - 180° = 035°
Solving bearing problems with trigonometry
Bearing problems often require you to work with angles that are not the bearing itself. You must identify the angle within the right-angled triangle you're using for calculation.
Key technique:
- Draw north lines at each point mentioned
- Mark the bearing as an angle from north (clockwise)
- Identify the right-angled triangle needed for calculation
- Find the angle within that triangle (often different from the bearing)
- Apply the appropriate trigonometric ratio
Example angle conversions:
- Bearing 050° means the angle from the east line is 90° - 50° = 40°
- Bearing 125° means the angle from the south line is 125° - 90° = 35°
- Bearing 310° means the angle from the north line is 360° - 310° = 50° (or from west: 310° - 270° = 40°)
Combined problems involving both concepts
CXC CSEC Mathematics Paper 2 frequently includes multi-step problems combining bearings with angles of elevation or depression. These questions test your ability to work with multiple triangles and apply the Pythagorean theorem alongside trigonometry.
Problem structure:
- A ship travels on a bearing from one port to another (creates one triangle)
- From the final position, an observer sees a lighthouse at an angle of elevation (creates a second triangle)
- You may need to find total distance, height, or a new bearing
Strategy:
- Solve the bearing/distance triangle first to establish horizontal positions
- Use those results as the base for the elevation/depression triangle
- Keep track of which measurements are horizontal and which are slant distances
- Apply Pythagoras when you need to find the hypotenuse of a right triangle
- Show all working clearly, as CXC awards method marks
Worked examples
Example 1: Angle of elevation (typical CSEC question)
Question: A tourist standing 45 metres from the base of the Pigeon Point lighthouse in Tobago observes the top of the lighthouse at an angle of elevation of 35°. Calculate the height of the lighthouse, giving your answer correct to one decimal place.
Solution: Draw a right-angled triangle with:
- Horizontal distance (adjacent) = 45 m
- Height of lighthouse (opposite) = h m
- Angle of elevation = 35°
Using tangent ratio: tan 35° = opposite/adjacent tan 35° = h/45 h = 45 × tan 35° h = 45 × 0.7002 h = 31.509 h ≈ 31.5 m
Marks awarded for: correct ratio selection (1 mark), correct substitution (1 mark), accurate calculation (1 mark).
Example 2: Angle of depression with distance calculation
Question: From the top of a 60-metre cliff at Negril, Jamaica, a coast guard officer observes a fishing boat at an angle of depression of 28°. Calculate the horizontal distance of the boat from the base of the cliff, correct to the nearest metre.
Solution: The angle of depression from the cliff equals the angle of elevation from the boat to the top of the cliff = 28°.
In the right-angled triangle:
- Vertical height (opposite) = 60 m
- Horizontal distance (adjacent) = d m
- Angle at the boat = 28°
Using tangent ratio: tan 28° = opposite/adjacent tan 28° = 60/d d = 60/tan 28° d = 60/0.5317 d = 112.85 d ≈ 113 m
Marks awarded for: diagram with angle correctly identified (1 mark), correct ratio (1 mark), correct rearrangement (1 mark), answer to required accuracy (1 mark).
Example 3: Bearing problem with two stages
Question: A cargo ship leaves Port of Spain on a bearing of 075° and travels 80 km to point P. From P, the ship then travels 65 km on a bearing of 165° to reach Point Q.
(a) Calculate the distance of Q east of Port of Spain, correct to 1 decimal place. (b) Calculate the distance of Q south of Port of Spain, correct to 1 decimal place.
Solution:
(a) Distance east:
First leg: Bearing 075° Angle from east line = 90° - 75° = 15° Eastward component = 80 × cos 15° = 80 × 0.9659 = 77.27 km
Second leg: Bearing 165° Angle from south line = 165° - 180° = -15° (or 15° east of south) Eastward component = 65 × sin 15° = 65 × 0.2588 = 16.82 km
Total eastward distance = 77.27 + 16.82 = 94.1 km
(b) Distance south:
First leg: Bearing 075° Northward component = 80 × sin 15° = 80 × 0.2588 = 20.70 km (north)
Second leg: Bearing 165° Southward component = 65 × cos 15° = 65 × 0.9659 = 62.78 km (south)
Net distance = 62.78 - 20.70 = 42.1 km south
Marks awarded for: correct angle identification (1 mark), correct use of trig ratios for each leg (2 marks), combining components correctly (2 marks), answers to required accuracy (1 mark).
Common mistakes and how to avoid them
Measuring angles from the vertical instead of horizontal — Always measure angles of elevation and depression from the horizontal line of sight. If you measure from the vertical, your entire calculation will be incorrect. Check your diagram shows a clear horizontal line.
Confusing angle of elevation with the angle in the calculation — The angle of depression from a cliff top is not always the angle you use in your triangle. Apply alternate angles: the angle of depression equals the angle of elevation from the opposite viewpoint. Mark both angles on your diagram.
Writing bearings with fewer than three figures — A bearing of 45° must be written as 045°. CXC mark schemes penalise incorrect notation. Always use three digits, padding with zeros when necessary.
Adding or subtracting the wrong amount for back bearings — Before calculating a back bearing, check whether the original bearing is greater or less than 180°. Add 180° when the bearing is less than 180°; subtract 180° when greater than 180°. A bearing of 200° going back gives 200° - 180° = 020°, not 380°.
Using the wrong trigonometric ratio — Identify which sides you know or need (opposite, adjacent, hypotenuse) relative to the marked angle before selecting sin, cos, or tan. Write "SOH-CAH-TOA" on your exam paper to remind yourself.
Calculator in radian mode — CSEC Mathematics uses degrees exclusively. Check your calculator displays "DEG" before starting calculations. A common error is getting nonsensical answers because the calculator is in radian or grad mode.
Exam technique for angles of elevation, depression and bearings
Command words: "Calculate" requires numerical answers with working shown; "determine" means find using appropriate methods; "hence" means use your previous answer in the next part. When you see "draw a diagram," marks are specifically allocated for an accurate, labelled sketch.
Diagram marks are easy marks — Even if you cannot complete the calculation, draw a clear, labelled diagram showing the scenario, the right angle, the given angle, and label sides with the given measurements. CXC typically awards 1-2 marks for correct diagrams.
Show the trigonometric ratio before substituting — Write "tan θ = opp/adj" or similar before inserting numbers. This demonstrates understanding and earns method marks even if your subsequent calculation contains errors. Method marks often comprise 60-70% of the total marks available.
State units and apply specified accuracy — If the question asks for "correct to 1 decimal place," an answer given to 2 decimal places or the nearest whole number loses the accuracy mark. Always include units (m, km, degrees) unless the question specifies otherwise.
Quick revision summary
Angles of elevation and depression are measured from the horizontal and form alternate angles in parallel-line scenarios. Use SOH-CAH-TOA to select the correct trigonometric ratio based on which sides are known or required. Three-figure bearings measure clockwise from north (000°-360°); calculate back bearings by adding or subtracting 180°. Always draw clear diagrams marking horizontal lines, north lines, and right angles. Multi-step problems require identifying separate triangles and combining results. Show your trigonometric ratio before substituting values, and ensure your calculator is in degree mode. State units and follow specified rounding instructions to secure all available marks.