HEIGHT AND DISTANCE

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:

• 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 .

           
          In this figure the line of sight is making an angle θ with the horizontal line. This angle is the angle of elevation.

• 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.

            
               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
                                     
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:

• 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.

SOME EXAMPLE AND SOLUTION
Example 1:The distance between two pillars of length 16 metres and 9 metres is x metres. If two angles of elevation of their respective top from the bottom of the other are complementary to each other, then the value of x (in metres) is
15
16
12 
9
SOLUTION: 
Given, AB = 16 m, CD = 9 m and BC = x metre
And, ∠ACB and ∠CBD are complementary.
∴  Let, ∠ACB = Θ and ∠CBD = (90° – Θ)
In ΔABC,

Example 2:  A point on a horizontal line through the base of a monument, the angle of elevation of the top of the monument is found to be such that its tangent is 1/5. On walking 138 metres towards the monument the secant of the angle of elevation is found to be √193/12. The height of the monument (in meter) is
35
49
42
56
SOLUTION: 
Given, the distance walking, CD = 138 m
Let, The height of the monument, AB = h metre
BD = x metre, ∠ACB = α and ∠ADB = β

⇒  7x = 12h
⇒  7(5h – 138) = 12h      [From eq. (i)]
⇒  35h – 966 = 12h
⇒  23h = 966
⇒  h = 42 m
∴  The height of the monument is 42 metre.

Example 3: The angle of elevation of the top of a tower from two points A and B lying on the horizontal through the foot of the tower are respectively 15° and 30°. If A and B are on the same side of the tower and AB = 48 metre, then the height of the tower is :
a) 24√3m
b) 24m 
c) 24√2m
d) 96m

SOLUTION: 

Example 4: If the angle of elevation of the sun changes from 30° to 45°, the length of the shadow of a pillar decreases by 20 metres. The height of the pillar is ? 
a) 20(√3 - 1)m
b) 20(√3 + 1)m
c) 10(√3 - 1)m
d) 10(√3 + 1) m
SOLUTION:

Let, the height of the pillar, AB = h metre.
When the sun's angle of elevation was 30°, then the length of shadow of the pillar is BD.
And, when the sun's angle of elevation is 45°, then the length of shadow of the pillar is BC = x metre (let).
When the sun changes from 30° to 45°, then the length of shadow of the pillar decreases CD = 20 (given)
∴  BD = BC + CD = (x + 20) m
In ΔABC,

Example 5: The top of two poles of height 24 m and 36 m are connected by a wire. If the wire makes an angle of 60° with the horizontal, then the length of the wire is ? 
a) 6m
b) 8 /√3m 
c) 8m
d) 6 /√3m