Application of Trigonometry

M A T H E M A T I C S A S S I G N M E N T – X
APPLICATION OF TRIGONOMETRY
Prepared By
Vishal Singh
Dhananjay Singh Bisht
C o n t a c t : 9 9 5 3 4 4 8 5 5 7

S T U D Y B O A R D
C O N T A C T : 9 9 5 3 4 4 8 5 5 7
Q1 : A tower stands vertically on the ground. The angle of elevation from a point on the ground, which is 30m
away from the foot of the tower is 30⁰. Find the height of the tower. [Ans: 17.32m]
Q2 : The angle of elevation of the top of a tower at a point on the ground, at a distance of 160m from its foot
is found to be 45⁰. Find the height of the tower. [Ans: 160m]
Q3 : A kite is flying with a string 150m long. It makes an angle of 60⁰ with the horizontal. Find the height of the
kite from the ground. [Ans: 129.9m]
Q4 : A kite is flying at a height of 60m above the ground. The string attached to the kite is temporarily tied to a
point on the ground. The inclination of the string to the ground is 60⁰, find the length of the string, assuming
that there is no slack in the string. [Ans: 69.28m]
Q5 : Find the angle of elevation of the top of a tower, which is 100√3 m high from a point at a distance of
100m from the foot of the tower on the horizontal plane. [Ans: 60⁰]
Q6 : A tree breaks due to strong wind and the broken part bends so that the top of the tree touches the
ground making an angle of 30⁰ with the ground . The distance between the foot of the tree to the point where
the top touches the ground is 8m. What was the height of the tree? [Ans: 8√3 m]
Q7 : The shadow of a tower , when the angle of elevation of the sun is 45⁰ is found to be 10m longer then
when it was 60⁰ . Find the height of the tower. [Ans: 23.66m]
Q8 : The shadow of a vertical pole is
1
√3
of its height. Show that sun’s elevation is 60⁰.
Q9 : The angle of elevation of a tower at a point is 45⁰. After going 40m towards the foot of the tower the
angle of elevation becomes 60⁰. Find the height of the tower. [Ans: 94.64m]
Q10 : The shadow of a tower standing on a level ground is found to be 45√3 m longer when the sun’s altitude
is 30⁰. Then when it was 60⁰. Find the height of the tower. [Ans: 67.5m]
Q11 : From a point on the ground , the angle of elevation of a building 10m high is 30⁰ . A flagstaff is fixed at
the top of the building and the angle of elevation of the top of the flagstaff from the same point is 45⁰. Find
the length of the flagstaff and the distance of the building from the point of observation.
[Ans: 7.32m; 17.32m]
Q12 : An aero plane at an altitude of 200m observes the angle of depression of opposite points on the two
banks of a river to be 45⁰ and 60⁰. Find in metres, the width of river. [Ans: 315.5m]
Q13 : A girl 1.6m tall stands at a distance of 3.2 from a lamp post and cast a shadow of 4.8m on the ground .
Find the height of the lamp post by using
a) Trigonometric ratios b) Properties of similar triangles
[Ans : 2.67m]

S T U D Y B O A R D
C O N T A C T : 9 9 5 3 4 4 8 5 5 7
Q14 : A person standing on the bank of a river observes that the angle of elevation of the top of a tree
standing on the opposite bank is 60⁰ . When he moves 40m away from the bank, he finds the angle to be 30⁰.
Find the height of the tree and the width of the river. [Ans: 34.64m]
Q15 : A pole 5m high is fixed on the top of the tower. The angle of elevation of the top of the pole observed
from a point A on the ground is 60⁰ and the angle of depression of the point A from the top of the tower is
45⁰. Find the height of the tower? [Ans : 6.83m]
Q16 : Two men on the either side of a temple 75m high observes the angle of elevation of the top of the
temple to be 30⁰ and 60⁰ respectively. Find the distance between the two men’s. [Ans : 173m]
Q17 : Kanchan is 1.5m tall. She is standing at a distance of 28.5m from a multi-storey building. The angle of
elevation of the top of the building from her eyes id 45⁰. Find the height of multi-storey building.
[Ans : 30 m]
Q18 : On the same side of the tower two objects are located. Observed from the top of the tower, their angle
of depression is 45⁰ and 60⁰. If the height of the tower is 150m, find the distance between the objects.
[Ans : 63.5 m]
Q19 : An aero plane when 3000m high, passes vertically above another aero plane at an instant when the angle
of elevation of the two planes from the same point on the ground are 60⁰ and 45⁰ respectively. Find the
vertical distances between the two planes. [Ans : 1268m]
Q20 : From the top of the building 60m high , the angle of depression of the top and the bottom of a tower
are observed to be 30⁰ and 60⁰ . Find the height of the tower. [Ans : 40m]
Q21 : The angle of depression of the top and the bottom of a building 8m high from the top of a multi-story
building are 30⁰ and 45⁰ respectively. Find the height of the multi-story building and the distance between two
building. [Ans : 18.92m ; 18.92m]
Q22 : From a point on a bridge across a river, the angle of depression of the banks on opposite sides of the
rivers are 30⁰ and 45⁰ . If the bridge is at a height of 30m from the banks, find the width of the river?
[Ans : 81.96m]
Q23 : There are two poles , one each on either bank of a river , just opposite to each other. One pole is 60m
high. From the top of this pole, the angle of depression of the top and the foot of the other pole are 30⁰ and
60⁰ respectively. Find the width of the river and the height of the other pole. [Ans : 34.64m ; 40m]
Q24 : A statue 1.6m tall stands on the top of a pedestal. From a point on the ground, the angle of elevation of
the top of the statue is 60⁰ and from the same point, the angle of elevation of the top of the pedestal is 45⁰.
Find the height of the pedestal. [Ans : 2.1856m]
Q25 : Two pillars of equal heights are on either side of a road which is 100m wide . At a point on the road
between the pillars, the angle of elevation of the top of the pillars are 60⁰ and 30⁰ respectively. Find the
position of the point between the pillars and the height of each pillar. [Ans : 25m ; 43.3m]

S T U D Y B O A R D
C O N T A C T : 9 9 5 3 4 4 8 5 5 7
Q26 : The horizontal distance between two towers is 80m. The angle of depression of the top of the first tower
when seen the top of the second tower is 30⁰. If the height of the second tower is 160m, find the height of the
first tower. [Ans : 113.8 m]
Q27 : A tree 12m high is broken by wind in such a way that its tip touches the ground makes an angle of 60⁰
with the ground . At what height at what height of from the ground, the tree is broken by the wind?
[Ans : 5.57m]
Q28 : A tree is broken by the storm in such a way that its tip touches the ground and makes an angle of 45⁰
with the grounds. Find the height of the tree if the horizontal distances of the tip from the bottom of tree is
10m. [Ans : 24.14m]
Q29 : The angle of elevation of the top of a hill at the foot of a tower is 60⁰ and the angle of elevation of the
top of the tower from the foot of hill is 30⁰. If the tower is 50m high, find the height of the hill. [Ans : 150m]
Q30 : At a point on the ground distant 15m from its foot , the angle of elevation of the top of the first storey of
a building is 30⁰. How high the second storey at the same point is 45⁰? [Ans : 6.34m]
Q31 : A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height ‘h’
metres. At a point on the plane, the angle of elevation of the bottom of the flagstaff is ‘a’ and that of top of the
flagstaff is ‘b’. Prove that the height of the tower is:
(ℎ Tan 𝑎 ) /(Tan 𝑏 − Tan 𝑎) (meter)
Q32 : A straight highway leads to the foot of a tower 50m high . From the top of tower, the angle of
depression of two cars standing on the highway are 30⁰ and 60⁰. Find the distances between the two cars and
how far is each car from the tower. [Ans : 57.67m ; 86.5m ; 28.3m]
Q33 : A man is standing on the deck of a ship which is 10m above the water level. He observes the angle of
elevation of the top of the hill as 60⁰ and the angle of depression of the base of the hill as 30⁰. Find the
distance of the hill from the ship and the height of the hill. [Ans : 17.32m ; 40m]
Q34 : The angle of elevation of the top of a building from the foot of a tower is 30⁰ and the angle of elevation
of the top of the tower from the foot of the building is 60⁰ if the tower is 50m high, find the height of the
building . [Ans : 16.67m]
Q35 : Two boats approach a lighthouse from opposite directions. The angle of elevation of the top of the
lighthouse from the boats are 30⁰ and 45⁰. If the distance between these boats are 100m, find the height of the
lighthouse. [Ans : 36.6m]
Q36 : From a window, 10m high above the ground of a house in a street the angles of elevation and
depression of the top and the foot of another house on the opposite side of the street are 60⁰ and 45⁰
respectively . Find the height of the opposite house. [Ans : 27.3m]

S T U D Y B O A R D
C O N T A C T : 9 9 5 3 4 4 8 5 5 7
Q37 : The angles of elevation and depression of the top and the bottom of a light house from the top of a
building 60m high are 30⁰ and 60⁰ respectively. Find:
a) The difference between the heights of the light house and the building.
b) Distance between light house and the building.
[Ans 20m; 34.64m]
Q38 : From the top of a hill, the angles of depression of two consecutive 1 Km milestones due east are found
to be 30⁰ and 45⁰. Find the height of the hill. [Ans : 1.366 Km]
Q39 : There is a small island in the middle of a river 100m wide and a tall tree stands on the islands. P and Q
are points directly opposite each other on the two banks and in line with the tree. If the angle of elevation of
the top of the tree from points P and Q are 30⁰ and 45⁰ respectively, find the height of the tree.
[Ans : 36.5m]
Q40 : From the top and foot of a tower 40m high, the angle of elevation of the top of a light house are found
to be 30⁰ and 60⁰ respectively. Find the height of the light house. Also find the distance of the top of the light
house from the foot of the tower. [Ans : 60m ; 69.28m]
Q41 : From a point on the ground how away from the foot of a tower, the angle of elevation of the top of the
tower and top of the water tank which is fixed at the top of tower are respectively 30⁰and 45⁰. Find the :
a) Height of the tower
b) Depth of the water tank
[Ans : 23.1m ; 16.9m]
Q42 : From the top of 7m high building, the angle of elevation of the top of a cable tower is 60⁰ and the angle
of depression of its foot is 45⁰. Find the height of the tower. [Ans : 19.124m]
Q43 : As observed from the top of a 75m high light house, the angles of depression of two ships are 30⁰ and
45⁰. If one ship is exactly behind the other on the same side of the light house, then find the distance between
the two ships. [Ans : 54.9m]
Q44 : The angle of elevation of an aero plane from a point on the ground is 45⁰. After a fight of 15 seconds, the
elevation changes to 30⁰. If the aero plane is flying at a constant height of 3000m, find the speed of the aero
plane. [Ans : 527.04 Km/hr]
Q45 : A man on the deck of a ship is 12m above water level. He observes that the angle of elevation of the top
of a cliff is 45⁰ and the angle of depression of the two base is 30⁰. Calculate the distance of the cliff from the
ship and the height of the cliff. [Ans : 32.784m ; 20.784m]
Q46 : A girl who is 1.2m tall spots a balloon moving with the wind in a horizontal line at a height of 88.2m from
the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is 60⁰. After some
time, the angle of elevation reduces to 30⁰. Find the distance travelled by the balloon during the interval.
[Ans : 58√3 m]

S T U D Y B O A R D
C O N T A C T : 9 9 5 3 4 4 8 5 5 7
Q47 : The angles of elevation of the top of a tower from two parts P and Q at distances of ‘a’ and ‘b’
respectively from the base and in the same straight line with it are complementary . Prove that the height of
the tower is (√a* √b).
Q48 : A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height 5m. At a
point on the plane, the angle of elevation of the bottom and the top of the flagstaff are 30⁰ and 60⁰
respectively. Find the height of the tower. [Ans : 2.5m]
Q49 : The measure of the angle of elevation of a cloud from a point 50m above a lake is 30⁰ and the measure
of the angle of depression of its reflection in the lake is 45⁰. Find the height of the cloud. [Ans : 186.6m]
Q50 : A man on the top of a tower on the sea shore find that a boat coming towards him takes 10 minutes to
the angle of depression to change from 30⁰ to 60⁰. How soon will the boat reach the sea shore? [Ans : 5 min]
Q51 : An aero plane flying horizontal 1 Km above the ground is observed at an elevation of 60⁰. After 10
seconds it’s elevation is observed to be 30⁰. Find the speed of the plane. [Ans : 415.66 Km/hr]
Q52 : The shadow of a flagstaff is three times as long as the shadow of the flagstaff when the sun rays meet
the ground at an angle of 60⁰. Find the angle between the sun rays and the ground at the time of longer
shadow. [Ans : 30⁰]
Q53 : The angle of elevation of the top of a vertical tower PQ , from a point X on the ground is 60⁰. At a point
Y, 40m vertically above X, the angle of elevation is 45⁰. Find the height of the tower PQ and the distance XQ.
[Ans : 109.3m ; 94.64m]
Q54 : At the foot of a mountain, the elevation of its summit is 45⁰ . After ascending 1 Km towards the mountain
up an incline of 30⁰. The angle of elevation changes to 60⁰. Find the height of the mountain.
[Ans : 1.366Km]

Application of Trigonometry

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