Angles of Elevation and Depression

When you look up at something, the angle of elevation is the angle between the horizontal and your line of sight. Here you stand twenty metres from a tree and look up to its top at forty five degrees. The ground, the tree and your line of sight make a right-angled triangle. From the forty five degree angle, the height h is opposite and twenty is adjacent, so tan forty five degrees equals h over twenty. Tan forty five degrees is one, so h is twenty metres.

Now you stand on top of a fifty metre cliff and look down at a boat. The angle of depression is measured down from the horizontal at your eye: thirty degrees. Draw a horizontal line at the top and the sea at the bottom. They are parallel, so the angle of depression at the top equals the angle of elevation at the boat. These are alternate angles. Now work at the boat: tan thirty degrees equals fifty over d, and tan thirty degrees is one over root three, so d is fifty root three metres.

Real people do not have their eyes on the ground. Here your eye is one and a half metres up, forty metres from a building, and the angle of elevation of the top is thirty five degrees. The horizontal starts at your eye, so the triangle only reaches the part of the building above eye level. Call that x. Tan thirty five degrees equals x over forty, so x is forty times zero point seven, which is twenty eight. Then add back the one and a half metres below your eye: the building is twenty nine point five metres tall.

Two people look at the same thirty metre tower. From A the angle of elevation is forty five degrees, and from B, further back, it is twenty seven degrees, with tan twenty seven degrees about zero point five. Each person makes their own right-angled triangle with the tower. In A's triangle, tan forty five degrees equals thirty over the distance, so A is thirty metres away. In B's triangle the distance is thirty divided by zero point five, which is sixty metres. They are on the same side, so the gap between them is sixty minus thirty, which is thirty metres.

From the top of a forty metre lighthouse, you see two boats on opposite sides. The angles of depression are forty five degrees to P and thirty degrees to Q. By alternate angles these are also the angles of elevation at the boats. For P, tan forty five degrees equals forty over the distance, so P is forty metres from the base. For Q, tan thirty degrees is one over root three, so Q is forty root three metres away. The boats are on opposite sides of the lighthouse, so add: they are forty plus forty root three metres apart. Quiz and exercises now.