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Practical use of Trigonometry Trigonometric knowledge can be used to estimate heights of objects or distances betweens points, for example, height of a mountain or breadth of a river. Exercise 1. A kite with a string 150 ft. makes an angle of 45° with the ground. Assuming that the string is straight, how high is the kite? 2. A tree 10 meters high casts a 17·3 meter shadow. Find the angle of elevation of the sun. 3. From the top of a hill, the angles of depression of two consecutive kilometre stones due east are 30 and 45 degrees. How high is the hill? 4. A 12 meter ladder is inclined to the vertical at angle 15°. How far is it from the base? 5. An observer in a lighthouse is 66 feet above the surface of the water. The observer sees a ship and finds the angle of depression to be 0·7°. Estimate the distance of the ship from the base of the lighthouse. Round the answer to the nearest 5 feet. 6. From the point on a ground level, you measure the angle of elevation to the top of a mountain to be 45°. Then you walk 200 m further away from the mountain and find that the angle of elevation is now 30°. Find the height of the mountain. Round the answer to the nearest meter. 7. A surveyor stands 30 yards from the base of a building. On top of the building is a vertical radio antenna. Let denote the angle of elevation when the surveyor sights to the top of the building. Let denote the angle of elevation when the surveyor sights to the top of the antenna. Express the length of the antenna in terms of the angles and . 8. A helicopter hovers 800 feet directly above a small island. From the helicopter, the pilot takes a sighting to a point directly ashore on the mainland, at the waters edge. If the angle of depression is 30°, how far off the coast is the island? 9. A ladder 18 feet long leans against a building. The ladder forms an angle of 60° with the ground. (i) How high up the side of the building does the ladder reach? (ii) Find the horizontal distance from the foot of the ladder to the base of the building. 10. Two satellite tracking stations, located at points A and B in a desert, are 200 miles apart. At a prearranged time, both stations measure the angle of elevation of a satellite as it crosses the vertical plane containing A and B. If the angles of elevation from A and from B are and respectively, express the altitude h of the satellite in terms of and . 11. From the base of a 30 m high building, the angle of elevation of a tower is 60°, and from the top of the building, it is 30°.Find the height of the tower. 12. A man on the top a building 30 meters high observes a man coming directly towards it at a uniform speed. If it takes 15 minutes for the angle of depression to change from 30° to 45°, how much time will it take after this for the man to reach the base of the building? Round your answer to nearest minute. 13. An aeroplane flying horizontally 1 km above the ground is observed at an elevation of 60°.If after 10 seconds, the elevation is observed to be 30°, find the uniform speed per hour of the plane. 14. A man standing south of a lamppost observes his shadow on the horizontal plane to be 24 ft long. On walking eastwards 300 ft, he finds his shadow as 30 ft. If his height is 6 ft, find the height of the lamp above the plane. 15. Two poles of equal height are standing opposite to each other on either side of a road, which is 30 ft wide. From a point between them on the road, the angles of elevation of the tops are 30° and 60°. Find the height of each pole, rounded to nearest feet. 16. A vertical tower is surmounted by a vertical flagstaff of height 10 ft. At a point on the plane, the angles of elevation of the bottom and the top of the flagstaff are and respectively. Prove that the height of the tower is 10 tan /(tan -tan ) feet. 17. An aeroplane when 6000 m high passes vertically above another plane at an instant when their angles of elevation at the same observing point are 60° and 45° respectively. How many meters higher is the one than the other? 18. At the foot of a mountain, the elevation of its peak is 45°. After ascending 100 m towards the mountain up a slope of 30° inclination, the elevation is found to be 60°. Find the height of the mountain (to the nearest meter). 19. A vertical pole is struck by a speeding car and breaks into two, the top striking the ground at an angle of 30° and at a distance of 10 feet from the foot of the pole. Find the total height of the pole. 20. The breadth of a street between two houses is 9 meters and the angle of depression of the top of one, as observed from the top of the other, which is 12 meters high, is 30°. Find the height of the other house. 21. A town B is 13 km south and 18 km west of town A. Find the bearing and distance of B from A. Answers 1. 75 2 ft 2. Approx 30° 3. 2·37 km 4. 3·10 m 5. 5400 ft 6. 273·22 m 7. 30 (tan -tan ) 8. 800 3m 9. (i) 9 3 ft (ii) 9 ft 10. 200 /(cot -cot ) miles 11. 45 m 12. 41 minutes 13. 240 3 km/hr 14. 106 ft 15. 13 feet 17. 2536 m 18. 137 m 19. 17·32 ft 20. 6·8 m 21. 22·2 km, S 54°10' W