Angles of elevation and depression

Any surface which is parallel to the surface of the earth is said to be horizontal. For example, the surface of liquid in a container is always horizontal, even if the container is held at an angle.

The floor of your classroom is horizontal. Any line drawn on horizontal surface is will also be horizontal. Any line or surface which is perpendicular to a surface is said to be vertical. The walls of your classroom are vertical. A plum-line is a mass which hangs freely on a thread.

1. Say whether the following are horizontal or vertical, or neither:
2. the table top
3. the door
4. the pictures
5. the floor boards
6. the back of the chair
7. the table legs
8. the ruler (on the table)
9. the line where the walls meet
10. the brush handle
11. the top edge of the small

Elevation and depression

Suppose that you are looking at an object in the distance.

If the object is above you, then the angle of elevation is the angle your eyes look up.

If the object is below you, the angle of depression is the angle your eyes look down.

Angles of elevation and depression are measured from the horizontal.

It is common mistake not to measure the angle of depression from the horizontal.

Using the angle of depression or elevation to an object, and knowing how far away the object is, enables us to find the height of the object using trigonometry.

The advantage of doing this is that it is very difficult to measure the height of a mountain or the depth of a canyon directly; it is much easier to measure how far away it is (horizontal distance) and to measure the angle of elevation or depression.

Suppose that we want to find the height of this tree.

We mark point A and measure how far it is from the base of the tree.

Then we measure the angle of elevation from A to the top of the tree.

Now,

h/x = tan(θ)

h = xtan(θ)

we have measured x and θ, so we can calculate tan(θ) and thus we can find h, which is the height of the tree.

Angles of elevation and depression can be measured with a simple instrument called a clinometers.

A plum-line

Example

Solve for x
Solution

Angle of depression = 34^o
But ∠ B = ∠ O (alternate angles)
therefore, ∠ B = 34^o
From triangle ABO, we have
tan 34^o = 40x
=> 0.6745 = 40x
=> 0.6745 x = 40
=> x = 400.6745
=> x = 59.30

ASSESSMENT

Which of these scenarios gives an angle of elevation or angle of depression;

1. A man is looking at a banner at the top of a building.
2. A woman looking at the coin on the floor.
3. A baby looking at an aeroplane in the sky.