Angle of Depression
The 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. It is measured from the horizontal down to the line of sight; if the horizontal distance is $d$ and the vertical drop is $h$, then $\tan(\text{depression angle}) = h/d = |\Delta y|/|\Delta x|$ 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
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. A lighthouse keeper $40$ m above sea level who sees a ship at a depression angle of $12^\circ$ finds the horizontal distance $= 40/\tan(12^\circ) \approx 188$ m. A standard instrument landing system glide slope of $3^\circ$ means an aircraft $5$ km from the runway sits at altitude $\approx 5000 \times \tan(3^\circ) \approx 262$ m above the runway threshold.
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, since they are alternate interior angles with a horizontal transversal, always equal and letting you redraw the figure from either observer's perspective. In geodesy, the zenith angle and the depression angle are complementary, and depression angles are used in LiDAR and sonar where sensors project downward beams and compute distances from return times and known angles.