NCERT Textbook: Some Applications in Trigonometry | Mathematics (Maths) Class 10 PDF Download

 Page 1


SOME APPLICATIONS OF TRIGONOMETRY 133
9
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 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 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
SOME APPLICA TIONS OF
TRIGONOMETR Y
2024-25
Page 2


SOME APPLICATIONS OF TRIGONOMETRY 133
9
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 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 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
SOME APPLICA TIONS OF
TRIGONOMETR Y
2024-25
134 MATHEMA TICS
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 CD of the minar
without actually measuring it, what information do you need? Y ou would need to know
the following:
(i) the distance DE at which the student is standing from the foot of the minar
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Page 3


SOME APPLICATIONS OF TRIGONOMETRY 133
9
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 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 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
SOME APPLICA TIONS OF
TRIGONOMETR Y
2024-25
134 MATHEMA TICS
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 CD of the minar
without actually measuring it, what information do you need? Y ou would need to know
the following:
(i) the distance DE at which the student is standing from the foot of the minar
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SOME APPLICATIONS OF TRIGONOMETRY 135
(ii) the angle of elevation, ? BAC, of the top of the minar
(iii) the height AE of the student.
Assuming that the above three conditions are known, how can we determine the
height of the minar?
In the figure, CD = CB + BD. Here, BD = AE, which is the height of the student.
To find BC, we will use trigonometric ratios of ? BAC or ? A.
In ? ABC, the side BC is the opposite side in relation to the known ? 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 A or cot A, as these ratios involve AB and BC.
Therefore, tan A = 
BC
AB
or cot A = 
AB
,
BC
 which on solving would give us BC.
By adding AE to BC, you will get the height of the minar.
Now let us explain the process, we have just discussed, by solving some problems.
Example 1 : A tower stands vertically on the ground. From a point on the ground,
which is 15 m away from the foot of the tower, the angle of elevation of the top of the
tower is found to be 60°. Find the height of the tower.
Solution : First let us draw a simple diagram to
represent the problem (see Fig. 9.4). Here AB
represents the tower, CB is the distance of the point
from the tower and ? ACB is the angle of elevation.
We need to determine the height of the tower, i.e.,
AB. Also, ACB is a triangle, right-angled at B.
To solve the problem, we choose the trigonometric
ratio tan 60° (or cot 60°), as the ratio involves AB
and BC.
Now, tan 60° =
AB
BC
i.e.,
3
 =
AB
15
i.e., AB = 15 3
Hence, the height of the tower is 15 3 m.
Fig. 9.4
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SOME APPLICATIONS OF TRIGONOMETRY 133
9
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 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 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
SOME APPLICA TIONS OF
TRIGONOMETR Y
2024-25
134 MATHEMA TICS
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 CD of the minar
without actually measuring it, what information do you need? Y ou would need to know
the following:
(i) the distance DE at which the student is standing from the foot of the minar
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SOME APPLICATIONS OF TRIGONOMETRY 135
(ii) the angle of elevation, ? BAC, of the top of the minar
(iii) the height AE of the student.
Assuming that the above three conditions are known, how can we determine the
height of the minar?
In the figure, CD = CB + BD. Here, BD = AE, which is the height of the student.
To find BC, we will use trigonometric ratios of ? BAC or ? A.
In ? ABC, the side BC is the opposite side in relation to the known ? 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 A or cot A, as these ratios involve AB and BC.
Therefore, tan A = 
BC
AB
or cot A = 
AB
,
BC
 which on solving would give us BC.
By adding AE to BC, you will get the height of the minar.
Now let us explain the process, we have just discussed, by solving some problems.
Example 1 : A tower stands vertically on the ground. From a point on the ground,
which is 15 m away from the foot of the tower, the angle of elevation of the top of the
tower is found to be 60°. Find the height of the tower.
Solution : First let us draw a simple diagram to
represent the problem (see Fig. 9.4). Here AB
represents the tower, CB is the distance of the point
from the tower and ? ACB is the angle of elevation.
We need to determine the height of the tower, i.e.,
AB. Also, ACB is a triangle, right-angled at B.
To solve the problem, we choose the trigonometric
ratio tan 60° (or cot 60°), as the ratio involves AB
and BC.
Now, tan 60° =
AB
BC
i.e.,
3
 =
AB
15
i.e., AB = 15 3
Hence, the height of the tower is 15 3 m.
Fig. 9.4
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136 MATHEMA TICS
Example 2 : An electrician has to repair an
electric fault on a pole of height 5 m. She needs
to reach a point 1.3m below the top of the pole
to undertake the repair work (see Fig. 9.5). What
should be the length of the ladder that she should
use which, when inclined at an angle of 60° to
the horizontal, would enable her to reach the
required position? Also, how far from the foot
of the pole should she place the foot of the
ladder? (You may take 
3
 = 1.73)
Solution : In Fig. 9.5, the electrician is required to
reach the point B on the pole AD.
So, BD = AD – AB = (5 – 1.3)m = 3.7 m.
Here, BC represents the ladder. We need to find
its length, i.e., the hypotenuse of the right triangle BDC.
Now, can you think which trigonometic ratio should we consider?
It should be sin 60°.
So,
BD
BC
 = sin 60° or
3.7
BC
 = 
3
2
Therefore, BC =
3.7 2
3
?
 = 4.28 m (approx.)
i.e., the length of the ladder should be 4.28 m.
Now,
DC
BD
 = cot 60° = 
1
3
i.e., DC =
3.7
3
 = 2.14 m (approx.)
Therefore, she should place the foot of the ladder at a distance of 2.14 m from
the pole.
Fig. 9.5
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SOME APPLICATIONS OF TRIGONOMETRY 133
9
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 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 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
SOME APPLICA TIONS OF
TRIGONOMETR Y
2024-25
134 MATHEMA TICS
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 CD of the minar
without actually measuring it, what information do you need? Y ou would need to know
the following:
(i) the distance DE at which the student is standing from the foot of the minar
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SOME APPLICATIONS OF TRIGONOMETRY 135
(ii) the angle of elevation, ? BAC, of the top of the minar
(iii) the height AE of the student.
Assuming that the above three conditions are known, how can we determine the
height of the minar?
In the figure, CD = CB + BD. Here, BD = AE, which is the height of the student.
To find BC, we will use trigonometric ratios of ? BAC or ? A.
In ? ABC, the side BC is the opposite side in relation to the known ? 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 A or cot A, as these ratios involve AB and BC.
Therefore, tan A = 
BC
AB
or cot A = 
AB
,
BC
 which on solving would give us BC.
By adding AE to BC, you will get the height of the minar.
Now let us explain the process, we have just discussed, by solving some problems.
Example 1 : A tower stands vertically on the ground. From a point on the ground,
which is 15 m away from the foot of the tower, the angle of elevation of the top of the
tower is found to be 60°. Find the height of the tower.
Solution : First let us draw a simple diagram to
represent the problem (see Fig. 9.4). Here AB
represents the tower, CB is the distance of the point
from the tower and ? ACB is the angle of elevation.
We need to determine the height of the tower, i.e.,
AB. Also, ACB is a triangle, right-angled at B.
To solve the problem, we choose the trigonometric
ratio tan 60° (or cot 60°), as the ratio involves AB
and BC.
Now, tan 60° =
AB
BC
i.e.,
3
 =
AB
15
i.e., AB = 15 3
Hence, the height of the tower is 15 3 m.
Fig. 9.4
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136 MATHEMA TICS
Example 2 : An electrician has to repair an
electric fault on a pole of height 5 m. She needs
to reach a point 1.3m below the top of the pole
to undertake the repair work (see Fig. 9.5). What
should be the length of the ladder that she should
use which, when inclined at an angle of 60° to
the horizontal, would enable her to reach the
required position? Also, how far from the foot
of the pole should she place the foot of the
ladder? (You may take 
3
 = 1.73)
Solution : In Fig. 9.5, the electrician is required to
reach the point B on the pole AD.
So, BD = AD – AB = (5 – 1.3)m = 3.7 m.
Here, BC represents the ladder. We need to find
its length, i.e., the hypotenuse of the right triangle BDC.
Now, can you think which trigonometic ratio should we consider?
It should be sin 60°.
So,
BD
BC
 = sin 60° or
3.7
BC
 = 
3
2
Therefore, BC =
3.7 2
3
?
 = 4.28 m (approx.)
i.e., the length of the ladder should be 4.28 m.
Now,
DC
BD
 = cot 60° = 
1
3
i.e., DC =
3.7
3
 = 2.14 m (approx.)
Therefore, she should place the foot of the ladder at a distance of 2.14 m from
the pole.
Fig. 9.5
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SOME APPLICATIONS OF TRIGONOMETRY 137
Example 3 : An observer 1.5 m tall is 28.5 m away
from a chimney. The angle of elevation of the top
of the chimney from her eyes is 45°. What is the
height of the chimney?
Solution : Here, AB is the chimney, CD the
observer and ? ADE the angle of elevation (see
Fig. 9.6). In this case, ADE is a triangle, right-angled
at E and we are required to find the height of the
chimney.
We have AB =  AE + BE = AE + 1.5
and DE =  CB = 28.5 m
To determine AE, we choose a trigonometric ratio, which involves both AE and
DE. Let us choose the tangent of the angle of elevation.
Now, tan 45° =
AE
DE
i.e., 1 =
AE
28.5
Therefore, AE = 28.5
So the height of the chimney (AB) = (28.5 + 1.5) m = 30 m.
Example 4 : From a point P on the ground the angle of elevation of the top of a 10 m
tall building is 30°. A flag is hoisted at the top of the building and the angle of elevation
of the top of the flagstaff from P is 45°. Find the length of the flagstaff and the
distance of the building from the point P. (You may take 
3
 = 1.732)
Solution : In Fig. 9.7, AB denotes the height of the building, BD the flagstaff and P
the given point. Note that there are two right triangles PAB and P AD. We are required
to find the length of the flagstaff, i.e., DB and the distance of the building from the
point P, i.e., PA.
Fig. 9.6
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