A man stands on the roof of a building so that his eyes are 132m above ground level. A trash can is on the ground 67.5m away from the building on the line directly beneath the man. What is the angle of depression of the man's line of sight to the trash can on the ground?
Sketch the right-angled triangle.
base = 67.5, height = 132
Draw his line of sight
by alternate angles in parallel lines, the angle of depression will be equal to the angle at the base of the triangle, so
tanØ = 132/67.5 = ...
Ø = 62.9°
To find the angle of depression, we need to use trigonometry.
First, let's draw a diagram to represent the situation. We have a building with a man standing on the roof, and a trash can on the ground below. The height of the man's eyes above the ground level is given as 132m, and the distance of the trash can from the building is given as 67.5m.
Now, let's define some variables:
- Let "A" be the position of the man's eyes on the roof.
- Let "C" be the position of the trash can on the ground.
- Let "B" be the projection of "A" on the ground, directly beneath the man's position.
The angle of depression can be defined as the angle "ACB." We need to find this angle.
We have a right-angled triangle "ABC" where:
- Side "AB" is the height of the building, which is 132m.
- Side "BC" is the distance from the building to the trash can, which is 67.5m.
To find the angle of depression, we can use the tangent function:
tan(angle) = opposite/adjacent.
In this case, the opposite side is "AB" and the adjacent side is "BC." Therefore, we have:
tan(angle) = AB / BC.
Substituting the values we know:
tan(angle) = 132 / 67.5.
Now, we can use a scientific calculator or an online calculator to find the arctan (inverse tangent) of this value, which will give us the angle.
arctan(tan(angle)) = arctan(132 / 67.5).
By evaluating this expression, we find that the angle of depression, to the nearest degree, is approximately 60 degrees.