## 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 60**

^{o}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. [ 30**

^{o}]**3. A tree is broken by the wind. The top stuck the ground at an angle of 30**

^{o}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 30**

^{o}. 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 60**

^{o}. When he was 40 m away from the bank he finds that the angle of elevation to be 30^{o}. 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 60**

^{o}and 45^{o}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 60**

^{o}and the angle of elevation of the top of tower from the foot of hill is 30^{o}. 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 30**

^{o}and 45^{o}, 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 60**

^{o }and 30^{o}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 45**

^{o}. After ascending 1 km towards the mountain up an inclination of 30^{o}, the elevation changes to 60^{o}. 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 30**

^{o}. From the bottom of the same building, the angle of elevation of the top of tower is found to be 30^{o}. 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 60**

^{o}to the road and Q observe that it is an angle of 45^{o}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 60**

^{o}. Find the angle between the sunrays and the ground at the time of long shadow. ( 30^{o})**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 45**

^{o.}Another boy is standing on the roof of 25 m high building and is flying his kite at an elevation of 30^{o.}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 30**

^{o}to 45^{o}. 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 45**

^{o}and 60^{o}. 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 60**

^{o}and 30^{o}. 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 60**

^{o}and the angle of depression of the base of the hill as 30^{o}.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 30**

^{o}and 60^{o}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 60**

^{o }and the angle of depression of the point ‘A’ from the top of tower is 45^{o}. Find the height of tower. [ 6.83m]**22. Man on a cliff observes a boat at an angle of depression of 30**

^{o}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 60^{o}. 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 30**

^{o}to 45^{o}, how soon after this will the car reach the observation tower. Give your answer correct to nearest seconds. [16 min. 24 sec.]

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