9. Some Applications of Trigonometry
Trigonometry
9. SOME APPLICATION OF TRIGONOMETRY
SECTION = A
Q1. The shadow of a 30 m high tower on the ground at some time of the day is 
m long , then the angle of elevation of the sun at that time is :
(A) 30° (B) 90° (C) 45° (D) 60°
Solution : (d) 60° .
[ Here , 
m , 
m
In 
we have ,
]
Q2. The ratio of the height of a tower and the length of its shadow on the ground is 
, then the angle of elevation of the sun is : [CBSE2017]
(a) 60° (b) 30° (c) 70° (d) 90°
Solution: (a) 60°
[ Here , 
In we have , 
Therefore, the angle of elevation of the sun is 60° . ]
Q3. The angle of the elevation of the top of a tower from a point on the ground ,which is 15m away from the foot of the tower, is 60° . The height of the tower is :[SEBA 2019]
(a) 15 m (b) 
m (c) 
m (d) 
m
Solution: (b) 
m
[ Here, 
m and 
In we have ,
m
Therefore, the height of the tower is m ]
Q4. The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower is 45° . The height of the tower is : [SEBA 2018]
(a) 30 m (b) 15 m (c) 10 m (d) 60 m
Solution: (a) 30 m
[ Here, 
m and 
In we have ,
m
Therefore, the height of the tower is 30 m . ]
Q5. A pole casts a shadow of length 
m on the ground , when the sun’s elevation is 60° then the height of the pole is : [CBSE2015]
(a) 6 m (b) 8 m (c) 12 m (d) 10 m
Solution: (A) 6 m
[ Here , 
and 
m
In we have ,
Therefore, the height of the pole is 6 m . ]
Q6. A ladder 15 m long makes an angle of 60° with the wall , then the height of the point where the ladder touches the wall is :
(a) 
m (b) 
m (c) 
m (d) 
m
Solution: (d) m
[ Here , 
and 
m
In we have , 
m
Therefore, the height of the pole is m .
Q7. In figure, AB is a 6 m high pole and CD is a ladder inclined at an angle of 60° to the horizontal and reaches up to a point D of pole . If AD = 2.54 m , then the length of the ladder is ( Use 
) : [CBSE2016]
(a) 3 m (b) 2 m (c) 4 m (d) 8 m
Solution: (c) 4 m
[ Here , 
m , 
m , 
m and 
In 
we have ,
m
Therefore, the length of the ladder is 4 m . ]
Q8. A ladder of length 
m reaches a window 15 m high , then the inclination of the ladder with the ground is :
(a) 30° (b) 45° (c) 60° (d) 90°
Solution: (b) 45° .
[ Here , 
m and 
m
In we have
Therefore, the inclination of the ladder with the ground is 45° . ]
Q9. The angle of depression of an object from the top of a tower of height 75 m is 30° .Then the distance of the object from the foot of the tower is : [SEBA 2017]
(a) 
m (b) 
m (c) 
m (d) 150 m
Solution: [ ]
Q10. If the angle of elevation of the sun is 45° , then the ratio between the tower and its shadow is : [ SEBA 2015 ]
(a) 
(b) 
(c) 
(d) 
Solution: (a) 1 : 1
[ let 
the height of the tower , 
the height of shadow and 
In we have ,
Therefore, the ratio between the tower and its shadow is 1 : 1 . ]
Q11. When the sun is 30° above the horizontal the length of the shadow cast by 50 m building is :
(a) m (b) 
m (c) m (d) 
m
Solution: (b) m
[ Here , 
and 
m
In 
we have ,
m
Therefore, the length of the building is m. ]
Q12. A ladder , leaning against a wall, makes an angle of 60° with the horizontal . If the foot of the ladder is 2.5 m away from the wall, then the length of the ladder is :
(a) 2.5 m (b) 5.2 m (c) 
m (d) 5 m
Solution: (d) 5 m
[ Here , 
m and 
In we have ,
m
Therefore , the length of the ladder is 5 m . ]
Q13. In figure, a tower AB is 20 m high and BC, its shadow on the ground , is 
m long , then the sun’s altitude is :
(a) 30° (b) 45° (c) 60° (d) 90°
Solution: (a) 30°
[ Here , 
m and 
m
In , we have 
]
Q14. A tower stands vertically on the ground . From a point on the ground, which is 15 m away from the foot the tower, the angle of elevation of the top of the tower is found to be 30° . The height of the tower is : [SEBA 2020]
(a) 
m (b) m (c) 15 m (d) m
Solution: (d) m
[ Here , 
m and 
In we have ,
m
Therefore, the height of the tower is m ]
Filled in the blanks
Q1. When an observer see an object situated in upward direction, the angle formed by line of sight with horizontal line is called angled of 
.
Solution: The angle of elevation .
Q2. When an observer see an object situated in downward direction, the angle formed by line of sight with horizontal line is called angled of .
Solution: The angle of depression .
Q3. The angle of elevation of the sun’s altitude when the height of the shadow of a vertical pole is equal to its height is 
.
Solution: 45°
[ Here , the height of the shadow and the height of the pole .
Given , 
In we have ,
Therefore, the angle of elevation is 45° . ]
Answers following the questions
Q1. In figure, the angle of elevation of the top of a tower AC from a point B on the ground is 60° . If the height of the tower is 20 m , find the distance of the point from the foot of the tower . [ CBSE 2020 Basic]
Solution: Here, 
m , 
and the distance of the point from the foot of the tower .
In we have , 
m
Therefore, the distance of the point from the foot of the tower is m .
Q2. In figure, the angle of elevation of the top of a tower from a point C on the ground, which is 30 m away from the foot of the tower is 30 . Find the height of the tower . [CBSE 2020 standard]
Solution: Here , m , 
And the height of tower .
In we have ,
m
Therefore, the height of the tower is m .
3. If the height of a vertical pole is 
times the length of its shadow on the ground, then find the angle of elevation of the sun at that time . [CBSE 2014]
Solution: Given, 
In we have ,
Therefore, the angle of elevation of the sun at that time is 60° .
SECTION = C
Q1. The shadow of a tower standing on a level ground is found to be 40 m longer when the sun's altiitude is 30° than when it is 60° . Find the height of the tower .
Solution: In figure, The height of the tower = AB , CD = 40 m ,
Angle of elevation , 
ADB = 30° and ACB = 60° ..
In ∆ABD we have ,
In ∆ABC we have ,
From (i) and (ii) , we get 
From (ii) we get , 
Therefore, height of the tower is 
.
Q2. From the top of a 7 m high building , the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foot is 45° . Determine the height of the tower .
Solution: In figure , Here, 
height of the building 
m ;
the distance between tower and building ;
and 
we have , 
In
we have , 
m 
From 
and 
we get , 
m
Therefore, the height of the tower is 
.
Q3. The angle of elevation of the top of a building from the foot of the tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60° . If the tower is 50 m high, find the height of the building .
Solution:
Q4. A statue , 1.6 m tall, stands on the top of a pedestal. From a point on te ground, the angle of elevation of the top of the statue is 60° and from the same point the angle of elevation of the top of the pedestal is 45° . Find the height of the pedestal .
SECTION = D
Q1. From a balloon vertically above a straight road , the angles of depression of two cars at an instant are found to be 45° and 60° . If the cars are 100 m apart , find the height of the balloon . [Examplar 2020 ]
Solution: let , 
(= PQ) be the height of the balloon .
In figure , AB = 100 m , PQ =
In 
, we have 
In 
, we have 
From and , we get 
From , we get 
m
Therefore , the height of the balloon is 
m .
Q2. 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. [Use 
1.73 ]
Solution: In figure, 
the height of the building 
m ;
the length of flagstaff .
∴ Angle of elevation are : 
and
In 
we have,
m
In
we have, 
m
m 
m
Therefore , the distance of the building from the point P is 
and the length of the flagstaff is 
.
Q3. A straight highway leads to the foot of a tower . A man standing at the top of the tower observes a car at an angle of depression of 30°, which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be 60°. Find the time taken by the car to reach the foot of the tower from this point.
Solution: let, 
m ; 
and 
[ Speed 
]
In 
we have , 
In 
we have , 
and we get , 
second
The time taken by the car to reach the foot of the tower is 3 second .
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