Angles of Elevation and Depression

Trigonometry isn't just for triangles on paper — it answers real "how high?" and "how far?" questions. The key is the line of sight from an observer to an object, and the angle it makes with the horizontal.

The angle of elevation is measured up from the horizontal — the angle you tilt your gaze to look up at a treetop or the top of a building.
The angle of depression is measured down from the horizontal — the angle you tilt your gaze to look down from a cliff or a window at something below.

When two people look at each other — you up at the cliff-top, someone down at the boat — the angle of elevation from below equals the angle of depression from above. The two horizontals are parallel, so these are alternate angles between parallel lines.

To solve a problem, draw a right-angled triangle: the horizontal distance, the vertical height, and the slanted line of sight. Then it's just SOH-CAH-TOA again.

For an observer looking at an object across a horizontal:

the angle of elevation is the angle measured up from the horizontal;
the angle of depression is the angle measured down from the horizontal;
between the same two points the two angles are equal (alternate angles between parallel horizontals);
model the situation with a right-angled triangle and solve with \sin, \cos or \tan.

A worked picture

Step through it: place the observer and the object, mark the angle of elevation, then read off the ratio that links the height to the distance.