In our day-to-day lives, we often encounter tall structures like towers, buildings, and trees, or even flying objects like kites and airplanes. Measuring their heights or our distance from them using a physical measuring tape is practically impossible.
Trigonometry provides us with a powerful mathematical tool to calculate these heights and distances easily. The core concept used for these real-world calculations relies on understanding the angle of elevation.
Now, let us learn some basic definitions.
Line of sight:
The line of sight is defined as the straight line drawn from the eye of an observer to the object viewed by the observer.
Horizontal line:
The straight horizontal line parallel to the ground that passes directly through the observer's eye.
Angle of elevation:
The angle of elevation is defined as the angle formed by the line of sight and the horizontal line when the point being viewed is above the horizontal level.
That is the case when we raise our heads to look at the objects.
Let us learn real-life situations based on the angle of elevation.
Example:
The altitude of the plane from point \(C\) on the ground is \(1500\ m\). If the angle of elevation of the person from point \(A\) is \(60^\circ\), then find the distance of the person from point \(A\) to \(C\).
Solution:
Let \(A\) be the position of the person, \(B\) be the position of the airplane, and \(C\) be the point.
Then, from the given data, we have:
\(\theta = 60^\circ\) and \(BC = 1500\ m\)
In the right triangle \(ACB\), we have:
\(tan\ \theta = \frac{BC}{AC}\)
\(\tan\ 60^\circ = \frac{1500}{AC}\)
\(AC = \frac{1500}{tan\ 60^\circ}\)
\(AC = \frac{1500}{\sqrt{3}}\)
\(AC = 500\sqrt{3}\)
Therefore, the distance between the points \(A\) and \(C\) is \(500\sqrt{3}\ m\).
Important!
- The object is always above the observer's eye level.
- The angle is always measured from the horizontal line, not from the ground.
- The angle of elevation lies between \(0^\circ\) and \(90^\circ\).