SCC Education

Application of Trigonometry

Application of Trigonometry
1 The angle of elevation of the top of a tower, from a point on the ground and at a distance of 150 m from its foot, is 30°. Find the height of the tower correct to one decimal place.

2.      From a point P on the level ground, the angle of elevation of the top of a tower is 30°. If the tower is 100 m high, how far is P from the foot of tower?

3.      A kite is flying at a height of 75 meters from the level ground, attached to a string inclined at 60° to the horizontal. Find the length of the string to the nearest meter.

4.      If the length of a shadow cast by a pole be √3 times the length of the pole, find the angle of elevation of the sun.

5. The angle of elevation of a cloud from a point 200 meters above a lake is 30° and the angle of depression of its reflection in the lake is 60°. Find the height of the cloud.

6.  (a) From a boat 300 meters away from a vertical cliff, the angles of elevation of the top and the foot of a vertical concrete pillar at the edge of the cliff are 55° 40' and 54°20' respectively. Find the height of the pillar correct to the nearest meter.

(b) From a man M, the angle of elevation of the top of a tree is 44°. What is the angle of elevation from the man of a bird perched half way up the tree?

7.      The upper part of a tree broken by wind, falls to the ground without being detached. The top of the broken part touches the ground at an angle of 38° 30' at a point 6 m from the foot of the tree. Calculate

(i) the height at which the tree is broken.
(ii) the original height of the tree correct to two decimal places.

8.      The angle of elevation of the top of a tower from a point A (on the ground) is 30°. On walking 50 m towards the tower, the angle of elevation is found to be 60°. Calculate

(i) the height of the tower (correct to one decimal place).
(ii) the distance of the tower from A.

9.      From the top of a church spire 96 m high the angles of depression of two vehicles on a road, at the same level as the base of the spire and on the same side of it are x° and y°, where tan x° = 1/4 and tan y° = 1/7. Calculate the distance between the vehicles.

10.  The shadow of a vertical tower on level ground increases by 10 m, when the altitude of the sun changes from 45° to 30°. Find the height of the tower correct to one decimal place.

Answers    1. 86·6 meters     2. 173·2 m                   3. 86·6 m      4. 30°      5. 400 m    6. (a) 21 m (b) 25° 46'      7. (i) 4·77 m (ii) 12·44 m    8. (i) 43·3 m (ii) 75 m  9. 288 m      10. 13·7 m

1.The angle of elevation of a ladder leaning against a wall is 60o and the foot of the ladder is 9.5 meter away from  the wall. Find the length of the ladder. [ 19m ]

2. If the length of the shadow cast by a pole be  times the length of the pole, find the angle of elevation of the sun. [ 30o ]
3.   A tree is broken by the wind. The top stuck the ground at an angle of 30o and at a distance of 30 m from the root.   Find the total height of the tree.
4.     A circus artist is climbing from the ground along a rope stretched from the top of vertical pole and tied at the ground level 30o. Calculate the distance covered by the artist in climbing to the top of the pole. [ 24 m ]
5. 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 60o. When he was 40 m away from the bank he finds that the angle of elevation to be 30o.  Find: -
(i) The height of the tee,
(ii) The width of the river, correct up to two decimal places.[(i)34.64m (ii) 20m]
6.  An aeroplane when flying at a height of 4000 m from the ground passes vertically above another aeroplane at an instant when the  angles of elevations of two planes to a same point on the ground are 60o and 45o respectively.  Find the vertical distance between the aeroplanes at that at that instant. [ 1693.34 m ]
7.  The angle of elevation of the top of the hill at the foot of a tower is 60o and the angle of elevation of the top of tower from the foot of hill is 30o. If the tower is 50 m high, what is the height of the hill. [ 150 m ]
8.    There is a small island in the middle of a 100 m wide river and a tall tree stands on the island. Let P and Q be       points directly opposite each other on the two banks, and in line with the tree. If the angles of elevation of the top     the tree from P and Q respectively are 30o and 45o, find the height of the tree.
9.        Two pillars of equal heights are on either sides of a roadway, which is 150 m wide. The angles of elevation of       the top of pillars are 60o  and 30o at a point on the roadway between the pillars. Find the position of the point    between the pillars and the height of each pillar.       (64.95m)

10.        At the foot of mountain, the elevation of its peak is 45o. After ascending 1 km towards the mountain up an inclination of 30o, the elevation changes to 60o. Find the height of mountain. (1.366 km)
11.        From the top of the building 15m high, the angle of elevation of the top of a tower is found to be 30o. From the       bottom of the same building, the angle of elevation of the top of tower is found to be 30o. Find the height of the     tower and the distance between the tower and the building. (22.5m, 12.975m)
12.        A fire in a building B is reported on the telephone to two fire stations P and Q, 10 km apart from each other on a    straight road. P observes that the fire is at angle of 60o to the road and Q observe that it is an angle of 45o to the       road. Which station should send its team and how much this team has to travel? (P, 7.32km)
13.        The shadow of a flagstaff is three times as long as the shadow of the flagstaff when the sunrays meet the ground  at an angle of 60o. Find the angle between the sunrays and the ground at the time of long shadow. ( 30o)
14.        From a point in the cricket ground, the angle of elevation of a vertical tower is found to be θ at a distance of  200m from the tower. On walking 125 m towards the tower the angle of elevation becomes 2θ. Find the height of   tower. (100m)
15.        A boy standing on the ground and flying a kite with 75 m of string at an elevation of 45o. Another boy is standing on the roof of 25 m high building and is flying his kite at an elevation of 30o. Both the boys are on the opposite side of the two kites. Find the length of the string that the second boy must have, so that the kites meet.(56.05 m)
16.        As observed from the top of light house, 100m high above the sea level, the angle of depression of a ship, sailing directly towards it, changes from 30o to 45o. Determine the distance traveled by the ship during the period of observation.  ( 73.2m)
17.        An aeroplane at an altitude of 200 m observes the angle of depression of opposite points on two banks of a river  to be 45o and 60o. Find the width of the river.  ( 315.4m)
18.        From the top of a cliff 150m high, the angles of depression of two boats are 60o and 30o. Find the distance  between the boats, if the  boats are (i) on the side of cliff. (ii) on the opposite sides of the cliff.     [ (i) 173.2m (ii) 346.4m ]
19.        A man standing on the deck of a ship, which is 10m above the water level, observe the angle of elevation of the  top of a hill as 60o and the angle of depression of the base of the hill as 30o.Calculate the distance of the hill from the ship and the height of the hill.   [17.3m, 40m].
20.        The angle of elevation and depression of the top and the bottom of a light house from the top of the building, 60m high, are 30o and 60o respectively. Find (i) The difference between the heights of the light house and the  building (ii) Distance between the light house and the building. [ (i) 20m, (ii) 34.64m]
21.A pole 5m high is fixed on the top of a tower. The angle of elevation of the top of the pole observed from Point ‘A’ on the ground is 60and the angle of depression of the point ‘A’ from the top of tower is 45o. Find the height of tower.  [ 6.83m]
22.        Man on a cliff observes a boat at an angle of depression of 30o which is approaching the shore to the point immediately beneath the observer with a uniform speed. Six minutes later, the angle of depression of the boat is  found to be 60o. Find the time taken by the boat to reach the shore. [ 9 minutes]
23.  A man on the top of a vertical observation tower observes a car moving at a uniform speed coming directly       towards it. If it takes 12 minutes for the angle of depression to change from 30o to 45o, how soon after this will       the car reach the observation tower. Give your answer correct to nearest seconds.  [16 min. 24 sec.]