Angles of elevation and depression

The horizontal reference

An angle of elevation is measured upwards from the observer’s horizontal. An angle of depression is measured downwards from it. When the two horizontal lines are parallel, the corresponding elevation and depression angles are equal by alternate angles.

In a right triangle, \(\tan\theta=\dfrac{h}{d}\), where \(h\) is the vertical difference in height and \(d\) is the horizontal distance. The line of sight is the hypotenuse, not the horizontal distance.

If the observer’s eye is above level ground, the trigonometric calculation gives the height above eye level. Add the eye height only when finding the object’s total height above the same ground level.

The diagrams illustrate the geometry and are not necessarily drawn to scale.

Use degree mode. Keep full calculator precision for later parts; round only the final requested quantity. State units and check whether the result is a height, horizontal distance or line-of-sight distance.