Angle of Elevation and Angle of Depression
Simple Explanation
The angle of elevation is the angle measured upward from a horizontal line to a higher object. The angle of depression is the angle measured downward from a horizontal line to a lower object. Because the horizontal reference lines are parallel, the angle of elevation from one point always equals the angle of depression from the other (alternate angles).
Why Do We Need It?
These two angles are the standard setup for real-world height-and-distance problems — measuring things you cannot directly reach, like a building's height or a ship's distance from a cliff.
See It
An observer at ground level looking up at the top of a building, with the angle of elevation marked between the horizontal line of sight and the line to the building top
Formula
The Six Trigonometric Ratios
sinθ=opp/hyp, cosθ=adj/hyp, tanθ=opp/adj, cscθ=hyp/opp, secθ=hyp/adj, cotθ=adj/opp
For an acute angle θ in a right triangle, each trigonometric ratio is a fixed ratio of two of the triangle's three sides — three basic ratios, and their three reciprocals.
- opp
- — the length of the side opposite angle θ
- adj
- — the length of the side adjacent to angle θ (not the hypotenuse)
- hyp
- — the length of the hypotenuse, opposite the right angle
When to use it: Whenever you know an angle and one side of a right triangle and need another side, or know two sides and need the angle.
Worked Example
Find a height using the angle of elevation
From a point 40 m from the base of a tower, the angle of elevation to the top is 32°. Find the tower's height. (tan32° ≈ 0.625.)
Why Does This Work?
The observer's horizontal sightline and the ground are parallel, and the tower's vertical height forms a right angle with the ground — so the angle of elevation, the horizontal distance, and the tower's height together form a right triangle, exactly like any other right-triangle trigonometry problem.
Real-Life Example
Air traffic control tracking
An air traffic controller measures the angle of elevation to an aircraft and its known horizontal distance to estimate its altitude.
This is the same elevation-angle trigonometry, used to estimate real-time altitude from ground-based angle measurements.
Practice
From the top of a 50 m cliff, the angle of depression to a boat is 20°. Find the boat's horizontal distance from the base of the cliff. (tan20° ≈ 0.364. Round to the nearest metre.)
MediumCommon mistake
Measuring the angle of depression from the wrong reference line — it is always measured from the HORIZONTAL at the observer's eye level, not from the vertical or from the ground.
Quick Review
- Angle of elevation: upward from horizontal. Angle of depression: downward from horizontal.
- Angle of elevation from one point = angle of depression from the other (alternate angles, parallel horizontals).
- Both set up a right triangle solvable with the trigonometric ratios.