The topic of height and distance in trigonometry is an important topic in competitive examinations point of view. Generally we have seen the problems where the height of a building is given and then from the top of this building the angles of elevation or depression are given for another building and we have to find the height of the second building. In this article we will cover these type of problems.
There are certain terms associated with the heights and distances which are described as follows:
Angle of Elevation: Let us consider a situation where a person is standing on the ground at point ‘O’ and he is looking at an object which is at some height (above the level of his eye) say the top of the building (P). The line joining the eye of the person with the top of the building is called the line of sight. The angle made by the line of sight with the horizontal line is called Angle of Elevation.
In this figure the line of sight is making an angle Î¸ with the horizontal line. This angle is the angle of elevation.
Angle of elevation of P from O = AOP
Angle of Depression: Now let us take another situation where the person is standing at some height (O) with respect to the object he is seeing (P). In this case the line joining the line of sight of the man with the object is called the line of sight. The angle made by the line of sight with the horizontal line is called angle of depression.
In the above figure ‘Î¸’ is the angle of depression.
The questions on this topic require some basic knowledge of Trigonometry. We should be aware of the basic trigonometric ratios and their values.
Let us recall that the ratios of the sides of a right angled triangle are called trigonometric ratios. These are sine, cosine, tangent, cosecant, secant and cotangent.
Let the Î”ABC is a right angled triangle.
Then Sin Î¸ = Opp/Hyp = AB/CB
Cos Î¸ = Base/Hyp = AB/CB
Tan Î¸ = Opp/ Base = AB/AC
Cosec Î¸ = CB/AB
Sec Î¸ = AB/BC
Cot Î¸ = AC/AB
Then Sin Î¸ = Opp/Hyp = AB/CB
Cos Î¸ = Base/Hyp = AB/CB
Tan Î¸ = Opp/ Base = AB/AC
Cosec Î¸ = CB/AB
Sec Î¸ = AB/BC
Cot Î¸ = AC/AB
Also we should know the values of these trigonometric ratios of some common angles as given in the following table:
0°

30°

45°

60°

90°
 
Sin

0

1/2

1/√2

√3/2

1

Cos

1

√3/2

1/√2

1/2

0

Tan

0

1/√3

1

√3

Infinite

The values in the given table will be useful while solving the questions on height and distances.
After going through some examples we will learn how to measure the height and how to find the distance.