CBSE Class 10 Mathematics Applications of Trigonometry Assignment Set 08

Heights And Distances

Basic Concepts

In this chapter, we will see how trigonometry is used for finding the heights and distances of various objects without the actual measurement. For this we will be using the concept of angles. In heights and distance two types of angles are used

• Angle of elevation
• Angle of depression

Before discussing these two types of angles, we will discuss about, "Line of Sight".

Line of sight

[Diagram representation]:
\( D \) (object)
              / (Line of Sight)
\( E \) (eye of the observer) ——————————— \( F \) (Horizontal line)

It is the line drawn from the eye of the observer to the point in the object viewed by the observer, as shown in the above diagram.

Angle of elevation

It is the angle formed by the line of sight with the horizontal when it is above the horizontal level i.e. in this case we raise our head to look at the object.

[Diagram representation]:
\( D \) (object)
              / (Line of Sight)
\( E \) (eye) —— [Angle of elevation] ——— \( F \) (Horizontal line)

Angle of depression

It is an angle formed by the line of sight with the horizontal when it is below the horizontal level i.e. in this case we lower our head to look at the object. as shown in the following diagram.

[Diagram representation]:
\( F \) (Horizontal level) ——— [Angle of depression] ——— \( E \) (Eye)
                                                                       / (Line of Sight)
\( D \) (object) —————————————————————————————— \( G \) (ground)

From the above working we noticed that the angle of elevation and angle of depression are alternate angles, which are equal

\[ \therefore \text{Angle of elevation} = \text{Angle of depression} \]

Note

• The height or length of an object or the distance between two distinct objects can be determined with the help of trigonometrical ratios.
• While solving a problem on heights and distances, generally one angle and one side or only two sides are given. Establish a relation in terms of t-ratios and solve it and get the required answer.