In today’s video presentation, we will start by thoroughly examining some vital aspects of height measurements and distance metrics before we delve into the core subject matter.
Height, as defined, refers to the vertical measurement of an object, while distance is simply the horizontal space that separates two distinct points.
Understanding Elevation Angle
Consider a scenario where an individual stands on the ground and gazes at an object located at a certain height—let's say, the top of a building. In this situation, the line of sight is a direct path that connects the observer's eye to the apex of the structure. The elevation angle can be derived from the line of sight in conjunction with the horizontal line extending from the observer.
Exploring the Depression Angle
Conversely, imagine a person positioned at a certain height who is looking down at an object, such as the base of the same building. Here, the line of sight refers to the trajectory from the observer's eye to the bottom of the building. The angle of depression is formed by the line of sight and the horizontal reach extending outward from the individual's viewpoint.
Let’s say the height of this building is 75 meters, and two individuals are situated at points A and B.
Assume distances AD = xm and DB = ym.
In the right-angled triangle Δ ADC, we have CD/AD = tan 30°
= 75/x = 1/√3, which simplifies to x = 75√3m.
Now, applying the same principle in right-angled triangle Δ CDB, we set CD/BD = tan 60°
75/y = √3 implies √3y = 75; thus y = 75/√3. Simplifying this gives y = 75/√3 × √3/√3 = 75√3/3 = 25√3.
Combining both distances, we find the total distance x + y = 75√3 + 25√3 = 100√3m.