TRIGONOMETRY
1. A man is walking along a straight road. He notices the top of a tower subtending an angle A = 60o with the ground at the point where he is standing. If the height of the tower is h = 15 m, then what is the distance (in meters) of the man from the tower? Answer: 8.66
2. A little boy is flying a kite. The string of the kite makes an angle of 30o with the ground. If the height of the kite is h = 18 m, find the length (in meters) of the string that the boy has used. Answer: 36
3. Two towers face each other separated by a distance d = 20 m. As seen from the top of the first tower, the angle of depression of the second tower's base is 60o and that of the top is 30o. What is the height (in meters) of the second tower? Answer: 23.09
4. A ship of height h = 24 m is sighted from a lighthouse. From the top of the lighthouse, the angle of depression to the top of the mast and the base of the ship equal 30o and 45o respectively. How far is the ship from the lighthouse (in meters)? Answer: 56.78
5. Two men on opposite sides of a TV tower of height 22 m notice the angle of elevation of the top of this tower to be 45o and 60o respectively. Find the distance (in meters) between the two men. Answer: 34.70
6. Two men on the same side of a tall building notice the angle of elevation to the top of the building to be 30o and 60o respectively. If the height of the building is known to be h =110 m, find the distance (in meters) between the two men.
Answer: 127.02
7. A pole of height h = 30 ft has a shadow of length l = 30.00 ft at a particular instant of time. Find the angle of elevation (in degrees) of the sun at this point of time. Answer: 45
8. You are stationed at a radar base and you observe an unidentified plane at an altitude h = 6000 m flying towards your radar base at an angle of elevation = 30o. After exactly one minute, your radar sweep reveals that the plane is now at an angle of elevation = 60o maintaining the same altitude. What is the speed (in m/s) of the plane? Answer: 115.47
OBJECTIVE QUESTIONS
1. Two ships are sailing in the sea on the two sides of a lighthouse. The angle of elevation of the top of the lighthouse is observed from the ships are 30º and 45º respectively. If the lighthouse is 100 m high, the distance between the two ships is: A. 173 m B. 200 m C. 273 m D. 300 m
2. A man standing at a point P is watching the top of a tower, which makes an angle of elevation of 30º with the man's eye. The man walks some distance towards the tower to watch its top and the angle of the elevation becomes 60º. What is the distance between the base of the tower and the point P? A. 43 units B. 8 units
C. 12 units D. Data inadequate E. None of these
3.The angle of elevation of a ladder leaning against a wall is 60º and the foot of the ladder is 4.6 m away from the wall. The length of the ladder is: A. 2.3 m B. 4.6 m C. 7.8 m D. 9.2 m
4.An observer 1.6 m tall is 203 away from a tower. The angle of elevation from his eye to the top of the tower is 30º. The heights of the tower is: A. 21.6 m B. 23.2 m C. 24.72 m D. None of these
5 From a point P on a level ground, the angle of elevation of the top tower is 30º. If the tower is 100 m high, the distance of point P from the foot of the tower is: A. 149 m B. 156 m C. 173 m D. 200 m
6. The angle of elevation of the sun, when the length of the shadow of a tree 3 times the height of the tree, is:
A. 30º B. 45º C. 60º D. 90º
7. The angle of elevation of the top of a tower 30 m high, from two points on the level ground on its opposite sides are 45 degrees and 60 degrees. What is the distance between the two points?
8. A spherical ball of radius 'r' placed on the ground subtends an angle of 600 at point A of the ground. What is the distance between the center of the ball and the point A?
9. 'a' and 'b' are the lengths of the base and height of a right angled triangle whose hypotenuse is 'h'. If the values of 'a' and 'b' are positive integers, which of the following cannot be a value of the square of the hypotenuse?
(1) 13 (2 ) 23 (3) 37 (4) 41
10. A man on the top of a vertical 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 0 to 45 0 , how soon after this, will the car reach the tower? Give your answer to the nearest second.
11. The hypotenuse of a right triangle is 1m less than twice the shortest side. If the third side is 1m more than the shortest side, find the sides and area of the triangle. Or
A swimming pool is filled with three pipes with uniform flow. The first two pipes operating simultaneously fill the pool in the same time during which the pool is filled by the third pipe alone. The second pipe fills the pool five hours faster than the first pipe and four hours slower than the third pipe. Find the time required by each pipe to fill the pool separately.
12.. A pole 5m high is fixed on the top of a tower. The angle of elevation of the top of the pole observed from a point ‘A’ on the ground is 60o and the angle of depression of the point ‘A’ from the top of the tower is 45o. Find the height of the tower.
13. 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 moves 30 m away from the bank, he finds the angle of elevation to be 30 o. Find the height of the tree and the width of the river.Or
The angle of elevation of a jet plane from a point A on the ground is 60 o .After flight of 7 seconds, the angle of elevation changes to 30 o . If the jet is flying at a constant height of 1750√3 m. find the speed of the jet plane.
14. 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 be the car reach the observation tower. 16.4 min appro Or
An aircraft is flying along a horizontal course AB directly towards an observer on the ground at P, maintaining an altitude of 5000 m. when the aircraft is at A, the angle of depression is 30 o and when at B ,it is 60 o respectively. Calculate the distance AB. 5773.33m
15. The horizontal distance between two towers is 140 m. The angle of elevation of the top of the first tower when seen from the top of the second tower is 30 O. If the height of the second tower is 60 m, find the height of the first tower.
16. Two stations due south of a leaning tower which leans towards north are at distances a & b from its foot. If angles of elevation of top of tower are α&β from these stations. Prove that inclination θ to the horizontal is given by cot Ø= (b cotα – a cotβ) / (b-a).
17. An airplane flying horizontally at a height of 1.5km above the ground is observed at a certain point on earth to subtend an angle of 60 o . After 15 seconds, its angle of elevation at the same point is observed to be 30°. Calculate the speed of the aero plane in km/h.
18. The angle of elevation of a cloud from a point 60 m above a lake is 30° and the angle of depression of the reflection of cloud in the lake is 60°. Find the height of the cloud.
19. The angle of elevation of a tower as seen from a point ‘A’ due North of it is ‘α’ and that as seen from a point B due East of A is ‘β’. Prove that the height of the tower is AB sinα sinβ/√sin2α – sin2β Or
The angle of elevation of a cloud from a point 60m above a lake is 300 and the angle of depression of the reflection of cloud in the lake is 600. Find the height of the cloud.
20. A man on a cliff observes a boat at an angle of depression of 30° 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°. Find the time taken by the boat to reach the shore Or
The angles of elevation of the top of a tower from two points 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 √ab
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 a point ‘A’ on the ground is 60 o and the angle of depression of the point ‘A’ from the top of the tower is 45 o. Find the height of the tower.
22. A man on the cliff observes a boat at an angle of depression of 30 approaching the shore immediately beneath the observer with uniform speed . Six minutes later the angle of depression of that boat is found to be 60 .Find the time taken by the boat to reach the shore . Or
A bird sitting on the top of a tree which is 80 m high . The angle of elevation of the bird from a point on the ground is 45. The bird flies away from the point of observation horizontally and remains constant height. After 2 seconds the angle of elevation of the bird from the point of observation becomes 30. Find the speed of the flying bird
23. The angle of elevation of a cloud from a point 200m above the lake is 300 and the angle of depression of its reflection in the lake is 60 0. Find the height of the cloud. Or
A man on a cliff observes a boat at angle of depression of 300 which is approaching the shore to the point immediately beneath the observer with uniform speed. Six minutes later, the angle of depression of the boat is found to be 600. Find the time taken by the boat to reach the shore.
24. A pole is projected outwards from a window 10 m above the ground of a building makes an angle of 300 with the wall. The angles of elevations of the bottom and top of the pole from a point on the ground are 300and 600 respectively. Find the length of the pole.
1. A man is walking along a straight road. He notices the top of a tower subtending an angle A = 60o with the ground at the point where he is standing. If the height of the tower is h = 15 m, then what is the distance (in meters) of the man from the tower? Answer: 8.66
2. A little boy is flying a kite. The string of the kite makes an angle of 30o with the ground. If the height of the kite is h = 18 m, find the length (in meters) of the string that the boy has used. Answer: 36
3. Two towers face each other separated by a distance d = 20 m. As seen from the top of the first tower, the angle of depression of the second tower's base is 60o and that of the top is 30o. What is the height (in meters) of the second tower? Answer: 23.09
4. A ship of height h = 24 m is sighted from a lighthouse. From the top of the lighthouse, the angle of depression to the top of the mast and the base of the ship equal 30o and 45o respectively. How far is the ship from the lighthouse (in meters)? Answer: 56.78
5. Two men on opposite sides of a TV tower of height 22 m notice the angle of elevation of the top of this tower to be 45o and 60o respectively. Find the distance (in meters) between the two men. Answer: 34.70
6. Two men on the same side of a tall building notice the angle of elevation to the top of the building to be 30o and 60o respectively. If the height of the building is known to be h =110 m, find the distance (in meters) between the two men.
Answer: 127.02
7. A pole of height h = 30 ft has a shadow of length l = 30.00 ft at a particular instant of time. Find the angle of elevation (in degrees) of the sun at this point of time. Answer: 45
8. You are stationed at a radar base and you observe an unidentified plane at an altitude h = 6000 m flying towards your radar base at an angle of elevation = 30o. After exactly one minute, your radar sweep reveals that the plane is now at an angle of elevation = 60o maintaining the same altitude. What is the speed (in m/s) of the plane? Answer: 115.47
OBJECTIVE QUESTIONS
1. Two ships are sailing in the sea on the two sides of a lighthouse. The angle of elevation of the top of the lighthouse is observed from the ships are 30º and 45º respectively. If the lighthouse is 100 m high, the distance between the two ships is: A. 173 m B. 200 m C. 273 m D. 300 m
2. A man standing at a point P is watching the top of a tower, which makes an angle of elevation of 30º with the man's eye. The man walks some distance towards the tower to watch its top and the angle of the elevation becomes 60º. What is the distance between the base of the tower and the point P? A. 43 units B. 8 units
C. 12 units D. Data inadequate E. None of these
3.The angle of elevation of a ladder leaning against a wall is 60º and the foot of the ladder is 4.6 m away from the wall. The length of the ladder is: A. 2.3 m B. 4.6 m C. 7.8 m D. 9.2 m
4.An observer 1.6 m tall is 203 away from a tower. The angle of elevation from his eye to the top of the tower is 30º. The heights of the tower is: A. 21.6 m B. 23.2 m C. 24.72 m D. None of these
5 From a point P on a level ground, the angle of elevation of the top tower is 30º. If the tower is 100 m high, the distance of point P from the foot of the tower is: A. 149 m B. 156 m C. 173 m D. 200 m
6. The angle of elevation of the sun, when the length of the shadow of a tree 3 times the height of the tree, is:
A. 30º B. 45º C. 60º D. 90º
7. The angle of elevation of the top of a tower 30 m high, from two points on the level ground on its opposite sides are 45 degrees and 60 degrees. What is the distance between the two points?
8. A spherical ball of radius 'r' placed on the ground subtends an angle of 600 at point A of the ground. What is the distance between the center of the ball and the point A?
9. 'a' and 'b' are the lengths of the base and height of a right angled triangle whose hypotenuse is 'h'. If the values of 'a' and 'b' are positive integers, which of the following cannot be a value of the square of the hypotenuse?
(1) 13 (2 ) 23 (3) 37 (4) 41
10. A man on the top of a vertical 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 0 to 45 0 , how soon after this, will the car reach the tower? Give your answer to the nearest second.
11. The hypotenuse of a right triangle is 1m less than twice the shortest side. If the third side is 1m more than the shortest side, find the sides and area of the triangle. Or
A swimming pool is filled with three pipes with uniform flow. The first two pipes operating simultaneously fill the pool in the same time during which the pool is filled by the third pipe alone. The second pipe fills the pool five hours faster than the first pipe and four hours slower than the third pipe. Find the time required by each pipe to fill the pool separately.
12.. A pole 5m high is fixed on the top of a tower. The angle of elevation of the top of the pole observed from a point ‘A’ on the ground is 60o and the angle of depression of the point ‘A’ from the top of the tower is 45o. Find the height of the tower.
13. 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 moves 30 m away from the bank, he finds the angle of elevation to be 30 o. Find the height of the tree and the width of the river.Or
The angle of elevation of a jet plane from a point A on the ground is 60 o .After flight of 7 seconds, the angle of elevation changes to 30 o . If the jet is flying at a constant height of 1750√3 m. find the speed of the jet plane.
14. 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 be the car reach the observation tower. 16.4 min appro Or
An aircraft is flying along a horizontal course AB directly towards an observer on the ground at P, maintaining an altitude of 5000 m. when the aircraft is at A, the angle of depression is 30 o and when at B ,it is 60 o respectively. Calculate the distance AB. 5773.33m
15. The horizontal distance between two towers is 140 m. The angle of elevation of the top of the first tower when seen from the top of the second tower is 30 O. If the height of the second tower is 60 m, find the height of the first tower.
16. Two stations due south of a leaning tower which leans towards north are at distances a & b from its foot. If angles of elevation of top of tower are α&β from these stations. Prove that inclination θ to the horizontal is given by cot Ø= (b cotα – a cotβ) / (b-a).
17. An airplane flying horizontally at a height of 1.5km above the ground is observed at a certain point on earth to subtend an angle of 60 o . After 15 seconds, its angle of elevation at the same point is observed to be 30°. Calculate the speed of the aero plane in km/h.
18. The angle of elevation of a cloud from a point 60 m above a lake is 30° and the angle of depression of the reflection of cloud in the lake is 60°. Find the height of the cloud.
19. The angle of elevation of a tower as seen from a point ‘A’ due North of it is ‘α’ and that as seen from a point B due East of A is ‘β’. Prove that the height of the tower is AB sinα sinβ/√sin2α – sin2β Or
The angle of elevation of a cloud from a point 60m above a lake is 300 and the angle of depression of the reflection of cloud in the lake is 600. Find the height of the cloud.
20. A man on a cliff observes a boat at an angle of depression of 30° 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°. Find the time taken by the boat to reach the shore Or
The angles of elevation of the top of a tower from two points 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 √ab
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 a point ‘A’ on the ground is 60 o and the angle of depression of the point ‘A’ from the top of the tower is 45 o. Find the height of the tower.
22. A man on the cliff observes a boat at an angle of depression of 30 approaching the shore immediately beneath the observer with uniform speed . Six minutes later the angle of depression of that boat is found to be 60 .Find the time taken by the boat to reach the shore . Or
A bird sitting on the top of a tree which is 80 m high . The angle of elevation of the bird from a point on the ground is 45. The bird flies away from the point of observation horizontally and remains constant height. After 2 seconds the angle of elevation of the bird from the point of observation becomes 30. Find the speed of the flying bird
23. The angle of elevation of a cloud from a point 200m above the lake is 300 and the angle of depression of its reflection in the lake is 60 0. Find the height of the cloud. Or
A man on a cliff observes a boat at angle of depression of 300 which is approaching the shore to the point immediately beneath the observer with uniform speed. Six minutes later, the angle of depression of the boat is found to be 600. Find the time taken by the boat to reach the shore.
24. A pole is projected outwards from a window 10 m above the ground of a building makes an angle of 300 with the wall. The angles of elevations of the bottom and top of the pole from a point on the ground are 300and 600 respectively. Find the length of the pole.
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