Heights and Distances: Theory & Concepts

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The topic of height and distance in trigonometry is an important topic from competitive examination point of view. Generally you must 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 you have to find the height of the second building. In this article we will cover these problems.
There are certain terms associated with the heights and distances which are described as follows:
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  1. Angle of Elevation: Let us consider a situation where a person is standing on the ground and he is looking at an object which is at some height say the top of the building. The line joining the eye of the man 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.
    Height And Distances Theory
    In this figure the line of sight is making an angle θ with the horizontal line. This angle is the angle of elevation.
  2. Angle of Depression: Now let us take another situation where the person is standing at some height with respect to the object he is seeing. In this case again the line joining the eye of the man with the bottom 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 depression.
    Height And Distances Theory
    In the above figure ‘θ’ is the angle of depression.
    Note: The angle of elevation is equal to the angle of depression.
The questions on this topic require basic knowledge of Trigonometry. You 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 ΔABC is a right angled triangle.
Height And Distances Theory
Then Sin θ = AB/AC
Cos θ = BC/AC
Tan θ = AB/BC
Cosec θ = AC/AB
Sec θ = AC/BC
Cot θ = BC/AB
Also you should know the values of these trigonometric ratios of some common angles as given in the following table:
  00 300 450 600 900
Sin 0 1/2 1/√2 √3/2 1
Cos 1 √3/2 1/√2 1/2 0
Tan 0 1/√3 1 √3 Not Defined
The values in the table will be used while solving the questions on height and distances. Let us now discuss some basic problems on height and distances. After going through these examples you will learn how to measure the height and how to find the distance.
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