Angle of Depression
TrigonometryThe angle of depression is the angle measured downward from a horizontal line to the line of sight toward an object below.
Formula
\tan(\theta) = \frac{\text{vertical drop}}{\text{horizontal distance}}
Definition
The angle of depression is how far you tilt your eyes downward from looking straight ahead to look at something below you, like a boat seen from a cliff.
Example
You stand on a cliff and look down $25^\circ$ below horizontal to see a boat in the water. That $25^\circ$ is the angle of depression.
Key Insight
The angle of depression from a high point to a low point equals the angle of elevation from that low point back up to you. They are alternate interior angles and are always equal.
Definition
The angle of depression is measured from the horizontal down to the line of sight to an object below the observer. If the horizontal distance is $d$ and the vertical drop is $h$, then $\tan(\text{depression angle}) = h/d$. The angle of depression from point A to point B equals the angle of elevation from B to A (alternate interior angles with a horizontal transversal).
Example
A lighthouse keeper $40$ m above sea level sees a ship at an angle of depression of $12^\circ$. Horizontal distance to ship $= 40 / \tan(12^\circ) \approx 40 / 0.213 \approx 188$ m.
Key Insight
The equality of angles of elevation and depression (as alternate interior angles) is a geometric property that simplifies many real-world problems. You can always redraw the figure from either observer's perspective.
Definition
The angle of depression $\theta$ is related to the slope of the line of sight by $\tan(\theta) = |\Delta y| / |\Delta x|$ for downward-sloping sight lines. In navigation and aviation, depression angles are used in terrain-following radar and in computing glide slopes, where the descent angle is the angle of depression from horizontal to the approach path.
Example
A standard instrument landing system (ILS) glide slope is $3^\circ$. For a horizontal distance of $5$ km from the runway, the aircraft's altitude above runway threshold $= 5000 \times \tan(3^\circ) \approx 5000 \times 0.0524 \approx 262$ m.
Key Insight
In geodesy, the zenith angle and the depression angle are complementary. Depression angles are used in LiDAR and sonar where sensors project downward beams and compute distances from return times and known angles.