Angles of Elevation, Depression and Bearings

Key Terms

Angle of elevation: Angle measured upward from the horizontal to a line of sight to an object above.
Angle of depression: Angle measured downward from horizontal; equal to the angle of elevation from the object by alternate interior angles.
True bearing: A direction measured clockwise from North, written as a 3-digit number from 000° to 360° (e.g. 045°, 270°).
Compass bearing: Uses cardinal directions: N45°E means 45° east of north (equivalent to bearing 045°).
Solving bearing problems: Draw a diagram with North lines at each point; identify angles using geometry; apply trig (right triangle or sine/cosine rule).
Co-interior angles: Parallel North lines cut by a transversal create co-interior (same-side) or alternate angles — use these to find unknown bearings.

Angles of Elevation and Depression

Both angles are measured from a horizontal line of sight. The horizontal is always the reference.

Angle of elevation: measured upward from horizontal to an object above.
Angle of depression: measured downward from horizontal to an object below.
Key fact: The angle of elevation from A to B equals the angle of depression from B to A (alternate angles on parallel horizontal lines).

Compass and True Bearings

Bearings describe direction of travel or direction to an object.

Compass bearings: N, NE, E, SE, S, SW, W, NW — 8 directions.

True bearings: Measured clockwise from North. Always written with 3 digits.
Examples: due North = 000°, due East = 090°, due South = 180°, due West = 270°

Components from bearing θ and distance d:
Northing = d cosθ | Easting = d sinθ

Worked Example 1 — Angle of Depression

Problem: A cliff is 60 m high. From the top, the angle of depression to a boat is 25°. Find the horizontal distance from the cliff base to the boat.

Step 1: Draw diagram. The horizontal from cliff top is parallel to sea level. Angle of depression = 25° below horizontal.

Step 2: In the right triangle formed: O = 60 m (cliff height), angle at top = 25°. Want A (horizontal distance).

Step 3: Use tan (O and A involved):

\[\tan 25° = \frac{O}{A} = \frac{60}{A}\] \[A = \frac{60}{\tan 25°} = \frac{60}{0.4663\ldots} \approx \mathbf{128.7 \text{ m}}\]

Answer: The boat is approximately 128.7 m from the base of the cliff.

Worked Example 2 — Bearing Components

Problem: A ship sails from point A on a bearing of 065° for 120 km to point B. Find the northing and easting components.

Step 1: The bearing 065° is measured clockwise from North. Draw a right triangle with the north direction as the adjacent side.

Step 2: The angle from North is 065°. Distance = 120 km.

\[\text{Northing} = 120\cos 65° = 120 \times 0.4226\ldots \approx \mathbf{50.7 \text{ km (North)}}\] \[\text{Easting} = 120\sin 65° = 120 \times 0.9063\ldots \approx \mathbf{108.8 \text{ km (East)}}\]

Answer: B is approximately 50.7 km north and 108.8 km east of A.

Hot Tip: Always draw a diagram for elevation/depression and bearing problems. Mark the horizontal line, the vertical, and the angle clearly. The right angle is formed between the horizontal and vertical, not between the slant and the vertical.

Drawing the Right Triangle

For elevation and depression problems, the horizontal line is always one side of your right triangle. The vertical (height) is another. The line of sight (or slope) is the hypotenuse.

Mark the angle of elevation/depression at the observer's position. Label:

Horizontal distance = A (adjacent to the angle)
Vertical height = O (opposite the angle)
Line of sight = H (hypotenuse)

Then apply SOH-CAH-TOA as usual.

Converting Between Bearing Systems

Compass	True Bearing	Compass	True Bearing
N 40° E	040°	S 40° E	140°
S 40° W	220°	N 40° W	320°

Rule: N x° E → true bearing = x°; S x° E → 180° − x°; S x° W → 180° + x°; N x° W → 360° − x°

Guide Example

Problem: A person on a cliff 90 m high sees a ship at an angle of depression of 20°. Find the slant distance from observer to ship.

Height (O) = 90 m, angle = 20°, want H (slant range).

\[\sin 20° = \frac{O}{H} = \frac{90}{H}\] \[H = \frac{90}{\sin 20°} = \frac{90}{0.3420\ldots} \approx 263.1 \text{ m}\]

Tip for Bearings: When a question asks how far north and how far east a ship has travelled, always draw a right-angled triangle with North pointing up, and use: north component = d cosθ, east component = d sinθ, where θ is the true bearing.

Mastery Practice

See Answers ➔

Fluency Find the height of a building if the angle of elevation from a point 40 m from its base is 52°. View Solution
Fluency From the top of a cliff 75 m high, the angle of depression to a boat is 18°. Find the horizontal distance from the base of the cliff to the boat. View Solution
Fluency A plane is flying at an altitude of 3000 m. It is observed at an angle of elevation of 14° from an observer on the ground. Find:
1. The horizontal distance from the observer to the point directly below the plane.
2. The actual (slant) distance from the observer to the plane.
View Solution
Understanding A surveyor at ground level sees the top of a tower at an angle of elevation of 34°. She then moves 20 m closer to the base of the tower and now sees the top at 49°. Find the height of the tower.
Hint: let h = height and d = initial distance. Write two equations using tan, then solve simultaneously.
View Solution
Understanding From the top of a lighthouse 45 m above sea level, the angles of depression to two boats on the same side are 24° (to the further boat) and 38° (to the closer boat). Find the distance between the two boats. View Solution
Understanding Convert the following bearings:
1. N 40° E to a true bearing.
2. S 65° W to a true bearing.
3. 125° to a compass bearing.
4. 305° to a compass bearing.
View Solution
Understanding A ship leaves port on a true bearing of 072° and sails 85 km. How far north and how far east of port is the ship? View Solution
Understanding Point B is 15 km from A on a bearing of 130°. Point C is 22 km from A on a bearing of 220°. Find:
1. Angle BAC (the angle at A between AB and AC).
2. The distance BC, using the cosine rule.
View Solution
Problem Solving Two observers A and B are 200 m apart on level ground. Observer A sees an aircraft at an angle of elevation of 42°. Observer B (on the same side as the aircraft) sees it at an angle of elevation of 58°. Find:
1. The horizontal distances from A and from B to the point directly below the aircraft.
2. The height of the aircraft.
3. Verify the height using both sets of measurements.
Hint: let h = height, d = horizontal distance from A to directly below plane, then (d − 200) = horizontal distance from B.
View Solution
Problem Solving A hiker walks 8 km due north, then 11 km on a bearing of 125°. Find:
1. The east component of the second leg.
2. The north component of the second leg.
3. The total northing (N/S displacement) from the starting point.
4. The total easting (E/W displacement) from the starting point.
5. The direct distance and bearing from the end point back to the start.
View Solution