BC is the distance between the tower’s foot and the building.
Tan is the trigonometric ratio involving sides AB, CD, BC, and angles B and C.
In ΔABC,
tan 60° = AB/BC
√3 = 50/BC
BC = 50/√3….(i)
In ΔBCD,
Tan 30° = CD/BC
1/√3 = CD/BC
1/√3 = 50/√3 [From (i)]
CD = 1/√3 × 50√3
CD = 50/3
Height of the building CD = 50/3 m.
Let’s take an example.
You and your friends have decided to go camping. You reach the top of a ridge while hiking and look down at the trail behind you. Your camp is visible in the distance. You wonder how far you have traveled and if there is any way to find out.
You can measure the angle between your horizontal line of sight and the camp using a small device known as a clinometer, and you realize the hill you just hiked up is 300 meters high. Could you use this information to determine how far your camp is away? (Assume the trail you hiked curves like a triangle’s side.)
Elevation and Depression angles
If you know an angle of elevation or a depressive one, you can use right triangles to find distances.
The diagram below depicts each of these angles.
The elevation angle is the point formed by the horizontal and upward lines of sight. For example, if you are standing on the ground and looking up at the peak of a mountain, you can calculate the elevation angle. The degree of depression is defined as the angle formed by the horizontal line of sight and the downward line toward an object. degree of depression is defined as the angle formed by the horizontal line of sight and the downward line of sight toward an object. For example, if you were standing on top of a hill or a building and looking down at something, you could measure the angle of depression. You can measure these angles using a clinometer or a theodolite. Clinometers and theodolites are there to measure the height of trees and other tall buildings.
We will solve several problems involving these angles and distances in this section.
Let the tower’s height be AB or the building’s height be CD.
The angle of elevation of the top of Building D from the foot of the Tower is 30 degrees, and the tip of the height of the top of Building A from the foot of Building C is 60 degrees.