Angles of elevation and depression

At the top of a lighthouse, a keeper looks down at a boat. At the same moment, someone in the boat looks up at the keeper. The keeper's gaze drops below level by one angle. The sailor's gaze rises above level by another. Which of those two angles is bigger has a clean answer, and it comes from one line drawn in the right place.

This is video four of eleven in Pythagoras, Trigonometry and Coordinate Geometry. The sums here are ordinary trigonometry, so if those feel shaky, Using trigonometry to find missing sides and angles comes first.

Some specifications list these two named angles only at Higher, or at Extended, so check your own paper; the trigonometry inside them is the same for everyone.

Both angles start from the same line: the horizontal, a flat line drawn level with the observer's eye. The observer is the person doing the looking. The second line is the line of sight, running straight from the observer's eye to the thing being looked at. When the observer looks up, the angle between the horizontal and the line of sight is the angle of elevation. Elevation means lifting, and the line of sight lifts above the horizontal. When the observer looks down, the angle between the horizontal and the line of sight is the angle of depression. Depression means pushing down, and the line of sight dips below. Both are measured from the horizontal at the observer's own eye, never from the ground and never from an upright wall. That is why the horizontal always gets drawn first, before the line of sight. Here is someone's diagram of a girl on a clifftop, looking at a boat below. What's wrong with where her angle of depression is drawn? It's measured from the cliff face, which is vertical. Depression is measured from the horizontal at her eye. The two names never depend on who is taller. Whoever looks up has an angle of elevation; whoever looks down has an angle of depression.

The lighthouse question needs two horizontals: one at the keeper's eye, and one at the boat, along the surface of the sea. Both lines are horizontal, so they are parallel. The line of sight cuts across both of them, making a Z shape. Suppose the keeper's angle of depression to the boat is fifteen degrees. The sailor's angle of elevation: fifteen, seventy-five or a hundred and five? And why? Fifteen. The two angles sit in the corners of that Z between parallel lines. They are alternate angles, and alternate angles are equal. The angle of elevation from the boat to the keeper equals the angle of depression from the keeper to the boat. That settles the lighthouse question: neither angle is bigger.

Elevation first, with numbers. A drone operator stands forty metres from the base of a tall crane, watching a parcel being lifted. The question treats the operator's eyes as level with the base of the crane. The horizontal goes in first, from the operator's eye along the ground to the crane. When the parcel is seventy-five metres up, the line of sight rises from the operator to the parcel. The angle of elevation sits at the operator, between the horizontal and that line. Which ratio links the forty and the seventy-five? Tan. Seventy-five is opposite and forty is adjacent, with no hypotenuse involved. Tan theta equals seventy-five over forty, which is one point eight seven five. An angle is wanted, so use the inverse. Press shift, tan, seventy-five, divide, forty, close the bracket, equals. The display reads sixty-one point nine two seven five one three zero six. To one decimal place, the angle of elevation from the operator to the parcel is sixty-one point nine degrees.

Depression works the same way from above. A lighthouse keeper stands sixty metres above sea level and spots a boat, and the angle of depression to it is fifteen degrees. The horizontal goes in first, out from the keeper's eye. The fifteen degrees sits below it, between the horizontal and the line of sight down to the boat. That fifteen degrees is outside the triangle. The alternate-angle fact moves it inside: the angle at the boat, between the sea and the line of sight, is also fifteen degrees. From the boat's fifteen degrees, which ratio links the sixty and the distance? Tan again. The sixty metres is opposite and the distance is adjacent, with no hypotenuse needed. Call the distance d. Tan fifteen equals sixty over d. This time d is on the bottom. Multiply both sides by d, then divide both sides by tan fifteen. That gives d equals sixty divided by tan fifteen. Press sixty, divide, tan, fifteen, close the bracket, equals. The display reads two hundred and twenty-three point nine two three zero four eight five. To one decimal place, the boat is two hundred and twenty-three point nine metres from the base of the lighthouse. Every angle faces its own side - look straight across. From the boat's angle of elevation, you look straight at the tower's height, and that is why the height is opposite.

The keeper and the sailor share one angle. Try the idea on some new views. From a window, you look down at a car. Elevation or depression, measured from where? Depression, measured down from the horizontal at your own eye. Here's a different one. A cat is at an elevation of twenty degrees. Its depression to you? Twenty degrees. The two horizontals are parallel, so the angles are alternate, and equal. Now a harder one: a balloon forty metres up, elevation forty-five degrees. How far along? Forty metres. Tan forty-five is one, so the opposite and the adjacent match, and the distance along the ground is the same as the height.

When this one feels done, mark it with a thumbs-up and cross it off your revision list. If the two angles still blur together, save it for later. Nobody sorts elevation from depression first time, and that's normal.

Next in the chapter: The sine rule and the area of a triangle.

For more, visit scholafly.com, or watch the next video.