What is the angle formed by the line of sight with the horizontal when the object viewed is below the horizontal level?

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.

Terms Related to Angle of Elevation

There are three terms related to angle of elevation and are given below: 

1. Line of sight

2. Horizontal line

3. Angle of elevation (image will be updated soon)

Line of Sight

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.

Formula for 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)

Comparison Between Angle of Elevation and Angle of Depression

Solved Examples:

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.

Heights and Distances

The line of sight is the line drawn from the eye of an observer to the point in the object viewed by the observer.

The angle of elevation of the point viewed is the angle formed by the line of sight with the horizontal when the point being viewed is above the horizontal level, i.e., the case when we raise our head to look at the object

Angle of depressionThe angle so formed by the line of sight with the horizontal is called the angle of depression.

Example: A tower stands vertically on the ground. From a point on the ground, which is 15 m away from the foot of the tower, the angle of elevation of the top of the tower is found to be 60°. Find the height of the tower.

Solution : First let us draw a simple diagram to represent the problem (see Fig. 9.4). Here AB represents the tower, CB is the distance of the point from the tower and ∠ ACB is the angle of elevation. We need to determine the height of the tower, i.e., AB. Also, ACB is a triangle, right-angled at B.

Example: An electrician has to repair an electric fault on a pole of height 5 m. She needs to reach a point 1.3m below the top of the pole to undertake the repair work (see Fig. 9.5). What should be the length of the ladder that she should use which, when inclined at an angle of 60° to the horizontal, would enable her to reach the required position? Also, how far from the foot of the pole should she place the foot of the ladder? (You may take 3 = 1.73)

Example: An observer 1.5 m tall is 28.5 m away from a chimney. The angle of elevation of the top of the chimney from her eyes is 45°. What is the height of the chimney?

Solution : Here, AB is the chimney, CD the observer and ∠ ADE the angle of elevation (see Fig. 9.6). In this case, ADE is a triangle, right-angled at E and we are required to find the height of the chimney.

To determine AE, we choose a trigonometric ratio, which involves both AE and DE. Let us choose the tangent of the angle of elevation.

Example: From a point P on the ground the angle of elevation of the top of a 10 m tall building is 30°. A flag is hoisted at the top of the building and the angle of elevation of the top of the flagstaff from P is 45°. Find the length of the flagstaff and the distance of the building from the point P. (You may take 3 = 1.732)

Solution : In Fig. 9.7, AB denotes the height of the building, BD the flagstaff and P the given point. Note that there are two right triangles PAB and PAD. We are required to find the length of the flagstaff, i.e., DB and the distance of the building from the point P, i.e., PA. Since, we know the height of the building AB, we will first consider the right ∆ PAB.

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