Heights and Distances

When you look at a faraway object, the straight line from your eye to that object is called the line of sight. If the object is higher than you, you raise your head and the line of sight makes an angle above the horizontal - this is the angle of elevation. If the object is lower than you, you lower your head and the line of sight makes an angle below the horizontal - this is the angle of depression. The horizontal is the imaginary level line through your eye. These three ideas - line of sight, elevation and depression - are the whole language of this chapter.

The trick is to draw a right-angled triangle. The vertical side is the height we want, the horizontal side is the distance along the ground, and the line of sight is the slanting side (the hypotenuse). The right angle is where the vertical meets the ground. Now the angle of elevation sits at the observer's position. Because tan(angle) = opposite over adjacent = height over distance, knowing the angle and one side immediately gives the other. Most problems use the standard angles 30, 45 and 60 degrees, because their tangents are clean values: tan 30 = 1/root3, tan 45 = 1, tan 60 = root3.

A person on the ground looking up at the top of a tower, and a person on top of the tower looking down at the first person, measure the SAME angle. This is because the horizontal at the eye and the horizontal at the ground are parallel, and the line of sight is a transversal cutting them - so the angle of elevation and the angle of depression are equal alternate angles. This single fact lets you redraw a depression problem as an elevation problem, which is often easier to solve.

Harder questions place two observers, or observe two objects. For example two people stand on opposite sides of a tower and each measures the angle of elevation of the top; or from a lighthouse the angles of depression of two ships are given. The method is always the same: form one right-angled triangle for each observer, write tan(angle) = height/distance for each, then add or subtract the horizontal distances to get the answer. The shared height (the tower or lighthouse) is the link between the two triangles.