Angles of Elevation & Depression
An angle of elevation looks up from horizontal; an angle of depression looks down. Both create right triangles perfect for SOH CAH TOA.
Introduction
These are the most common “word problem” applications of trigonometry. Nearly every real-world trig problem involves looking up at something (elevation) or down at something (depression) and forming a right triangle with the horizontal.
Past Knowledge
SOH CAH TOA (8.2.3–8.2.5). Inverse trig (8.3.1). Alternate interior angles.
Today's Goal
Draw diagrams and solve elevation/depression problems.
Future Success
Law of Sines (8.3.3), surveying, aviation.
Key Concepts
Definitions
- Angle of Elevation: The angle formed by a horizontal line and the line of sight upward to an object
- Angle of Depression: The angle formed by a horizontal line and the line of sight downward to an object
Key Property
The angle of elevation from point A to point B equals the angle of depression from B to A. Why? They are alternate interior angles formed by the line of sight and two parallel horizontal lines.
Problem-Solving Strategy
- Draw a diagram with a horizontal line and the line of sight
- Identify where the right angle is (usually at the base)
- Label the angle, opposite, adjacent, and/or hypotenuse
- Choose sin, cos, or tan and solve
Worked Examples
Elevation — Building Height
Standing 50 m from a building, the angle of elevation to the top is 62°. How tall is the building?
m
meters
Depression — Cliff to Boat
From a 120 ft cliff, the angle of depression to a boat is 25°. How far is the boat from the base of the cliff?
Depression angle = 25° → the angle at the boat is also 25° (alternate interior angles)
→ ft
feet
Double Observation
You observe a tower from two points along a straight line. From point A, the elevation is 30°. From point B (100 m closer), the elevation is 45°. Find the tower height.
Let = distance from B to the tower base. Then A is at .
From B: →
From A: →
Set equal:
→
m → m
meters
Common Pitfalls
Placing the Depression Angle Inside the Triangle
The angle of depression is measured from the horizontal line, not from the vertical. If depression = 25°, the angle inside the triangle (at the observer) is . Or use the alternate interior angle at the bottom which IS 25°.
Forgetting Eye Height
In real problems, the observer's eye height above ground must be added to or subtracted from the calculated height. A person measuring a building needs to add their 5.5-foot eye height.
Real-Life Applications
Aviation — Glide Slope
An airplane's instrument landing system uses a 3° glide slope. From 10 miles out, the plane is at miles (about 2,770 ft) — which is the correct approach altitude.
Search & Rescue — Spotting from Helicopters
A helicopter pilot at 500 ft sees a raft at a depression angle of 12°. The raft is ft away horizontally.
Practice Quiz
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