9.1 Heights and Distances

In the previous chapter, you have studied about trigonometric ratios. In this chapter, you will be studying about some ways in which trigonometry is used in the life around you.

Let us consider Fig. 8.1 of prvious chapter, which is redrawn below in Fig. 9.1.

Fig. 9.1

In this figure, the line $\mathrm{AC}$ drawn from the eye of the student to the top of the minar is called the line of sight. The student is looking at the top of the minar. The angle $\mathrm{BAC}$, so formed by the line of sight with the horizontal, is called the angle of elevation of the top of the minar from the eye of the student.

Thus, 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 (see Fig. 9.2).

Fig. 9.2

Now, consider the situation given in Fig. 8.2. The girl sitting on the balcony is looking down at a flower pot placed on a stair of the temple. In this case, the line of sight is below the horizontal level. The angle so formed by the line of sight with the horizontal is called the angle of depression.

Thus, the angle of depression of a point on the object being viewed is the angle formed by the line of sight with the horizontal when the point is below the horizontal level, i.e., the case when we lower our head to look at the point being viewed (see Fig. 9.3).

Fig. 9.3

Now, you may identify the lines of sight, and the angles so formed in Fig. 8.3. Are they angles of elevation or angles of depression?

Let us refer to Fig. 9.1 again. If you want to find the height $\mathrm{CD}$ of the minar without actually measuring it, what information do you need? You would need to know the following:

(i) the distance $\mathrm{DE}$ at which the student is standing from the foot of the minar (ii) the angle of elevation, $\mathrm{\angle }\mathrm{BAC}$, of the top of the minar

(iii) the height $\mathrm{AE}$ of the student.

Assuming that the above three conditions are known, how can we determine the height of the minar?

In the figure, $\mathrm{CD}=\mathrm{CB}+\mathrm{BD}$. Here, $\mathrm{BD}=\mathrm{AE}$, which is the height of the student.

To find $\mathrm{BC}$, we will use trigonometric ratios of $\mathrm{\angle }\mathrm{BAC}$ or $\mathrm{\angle }\mathrm{A}$.

In $\mathrm{△}\mathrm{ABC}$, the side $\mathrm{BC}$ is the opposite side in relation to the known $\mathrm{\angle }\mathrm{A}$. Now, which of the trigonometric ratios can we use? Which one of them has the two values that we have and the one we need to determine? Our search narrows down to using either $\tan \mathrm{A}$ or $\cot \mathrm{A}$, as these ratios involve $\mathrm{AB}$ and $\mathrm{BC}$.

Therefore, $\tan \mathrm{A}=\frac{\mathrm{BC}}{\mathrm{AB}}$ or $\cot \mathrm{A}=\frac{\mathrm{AB}}{\mathrm{BC}}$, which on solving would give us $\mathrm{BC}$.

By adding $\mathrm{AE}$ to $\mathrm{BC}$, you will get the height of the minar.

Now let us explain the process, we have just discussed, by solving some problems.

9.2 Summary

In this chapter, you have studied the following points :

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

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

(iii) The angle of depression of an object viewed, is the angle formed by the line of sight with the horizontal when it is below the horizontal level, i.e., the case when we lower our head to look at the object.

2. The height or length of an object or the distance between two distant objects can be determined with the help of trigonometric ratios.