The term ‘angle of elevation’ is a widely used concept when the application of trigonometry comes into the picture for finding the heights and distances of various objects without actually measuring them, using the trigonometric ratios like sin, cos and tan.
In this article, you will learn about the definition of angle of elevation, terms related to angle of elevation, its formula and comparison between angle of elevation and angle of depression along with some solved examples.
The term ‘angle of elevation’ is defined as the angle formed between the horizontal line and the oblique line from some object above the observer’s eye.
In other words, whenever you see an object above you, the angle formed between the horizontal line and the oblique line joining your eye and the object is called the angle of elevation. (image will be updated soon)
In the above figure, an observer is looking at an object which is above the horizontal line forming an angle ‘θ’ between the line of sight and the horizontal line. Here, if we join an imaginary perpendicular line from the object to the horizontal line, a right-angled triangle will be formed. Thus, we can now use the concepts of trigonometry to find the distance between observer and object or, the height at which the object is from the horizontal line.
There are three terms related to angle of elevation and are given below:
Line of sight
Horizontal line
Angle of elevation (image will be updated soon)
The line of sight is the oblique line drawn from the observer’s eye to the point being viewed on the object.
In the above figure, the oblique line AC is called the line of sight.
Horizontal Line
The imaginary line assumed for the horizontal eye level of the observer is here termed as horizontal line.
In the above figure, the line AB is the horizontal line.
When we raise our head to look at the object, the angle so formed by the line of sight with the horizontal line is called angle of elevation. It is measured in degrees and denoted by ‘θ’.
In the above figure, angle BAC is called the angle of elevation.
The formula for finding the angle of elevation depends on the information provided to us like, height of the object from horizontal level, horizontal distance of object from the observer and the oblique distance of the object from the observer. Based on this provided information, we can use the concept of trigonometric ratios to find the angle of elevation.
Let ‘θ’ be the angle of elevation then, the formula for angle of elevation if the height of object and its horizontal distance from the observer is given by:
Tan θ = (Height of the object from horizontal level)/(Horizontal distance of object form the observer)
Angle of elevation | Angle of depression |
When we see an object above the horizontal line, then the angle between line of sight and horizontal line is called angle of elevation. | When we see an object below the horizontal line, then the angle between line of sight and horizontal line is called angle of depression. |
It is the case when we raise our head to look at the object. | It is the case when we lower our head to look at the object. |
For example: if you are on ground and looking at the top of tower then, it is the case of angle of elevation. | For example: if you are in your balcony and looking down the flower pots on the ground then, it is the case of angle of depression. |
(image will be updated soon) | (image will be updated soon) |
Q.1. The angle of elevation of the top of a tower is 30° from a point on the ground, which is 15 m away from the foot of the tower. Find the height of the tower. (image will be updated soon)
Solution:
Let AB be the tower of height ‘h’ meters, and the angle of top of tower from a point ‘C’ on the ground is 30°.
In right angled triangle ABC,
Tan 30⁰ = (Height of the object from horizontal level)/(Horizontal distance of object form the observer)
Or, Tan 30⁰= \[\frac{h}{15 m}\] (∵ Tan 30° =\[\frac{1}{\sqrt 3}\])
Or, \[\frac{1}{\sqrt 3}\] = \[\frac{h}{15 m}\]
Or, h = \[\frac{15 m}{\sqrt 3}\]
Or, h = 5 \[\sqrt 3\] m = 5 × 1.732 = 8.66 m
Therefore, the height of the tower is 8.66 m.
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