Trigonometry in the Right Triangle · Grade 10
What is the difference between an angle of elevation and an angle of depression?
Both are measured from the horizontal. An angle of elevation is the angle you raise your line of sight through to look at something higher; an angle of depression is the angle you lower it through to look at something lower. Between two observers looking at each other the two angles are equal, because they are alternate angles between parallel horizontals.
Learning objectives
- Draw a diagram for a word problem and mark on it the angle of elevation or of depression
- Calculate the height of a building or a tree from a given distance and angle
- Solve a problem that needs two right triangles sharing a common side
- State the answer with the units and the accuracy the question asks for
Both from the horizontal
The first mistake in any question of this kind is measuring from the wall or the pole. Both angles are measured from the horizontal line through the eye, never from the vertical.
An angle of elevation appears when you look up — at the top of a building, at an aeroplane. An angle of depression appears when you look down — from a roof at a person in the street.
Why they are equal
The horizontal line of the observer above is parallel to the horizontal line of the observer below. The line of sight cuts across both, and alternate angles between parallels are equal.
This is very useful: a question giving an angle of depression from a roof can be solved exactly as though it gave an angle of elevation from the ground, which saves drawing a second diagram.
Setting the question up
Always draw. Mark the horizontal, the line of sight, and the right triangle they form with the height. Most questions then turn into an ordinary side-and-angle exercise.
Watch for eye height. If the question gives it, the height you calculate is above eye level and must be added at the end — and this is the point that trips up more word problems than any other.
Worked examples
A person stands 20 metres from a building and sees its top at an angle of elevation of 40 degrees. How high is the building above their eye level?
- The known side is the adjacent leg and the wanted one is the opposite, so tangent
- Tangent 40 equals the height over 20
- Tangent 40 is about 0.839
- Multiply by 20
Answer: About 16.8 metres above eye level
If their eye height is 1.6 metres, how tall is the building?
- The tangent gave the height above the line of sight only: 16.8 metres
- A building's height is measured from the ground, so add the eye height
- Add: 16.8 plus 1.6
Answer: About 18.4 metres
From a roof 30 metres high a car is seen at an angle of depression of 25 degrees. What is the horizontal distance to it?
- The angle of depression equals the angle of elevation from the car
- Tangent 25 equals 30 over the distance
- Rearrange: the distance equals 30 over tangent 25
- Tangent 25 is about 0.466
Answer: About 64.3 metres
Common mistakes
- Measuring the angle from the vertical instead of the horizontal
- Both angles are defined from the horizontal line. Measuring from the wall gives the complementary angle, and the answer is wrong by exactly that difference.
- Forgetting to add the eye height
- Trigonometry gives the height above the line of sight only. The question almost always asks for the height from the ground, which means adding it back.
- Assuming depression is larger than elevation
- Between two observers looking at each other they are equal, being alternate angles between parallel horizontals. There is no need to calculate them separately.
What to remember
- Both angles are measured from the horizontal, not the vertical.
- Elevation and depression between two observers are equal.
- Always draw before calculating.
- Eye height is added at the end.