(1) Determine whether the following measurements produce one triangle, two triangles or no triangle:
∠B = 88°, a = 23, b = 2. Solve if solution exists.
(2) If the sides of a triangle ABC are a = 4, b = 6 and c = 8, then show that 4 cosB + 3cosC = 2. Solution
(3) In a triangle ABC, if a = √3 − 1, b = √3 + 1 and C = 60°, find the other side and other two angles Solution
(4) In any triangle ABC, prove that the area triangle = b^{2} + c^{2} − a^{2}/4 cotA. Solution
(5) In a triangle ABC, if a = 12 cm, b = 8 cm and C = 30°, then show that its area is 24 sq.cm. Solution
(6) In a triangle ABC, if a = 18 cm, b = 24 cm and c = 30 cm, then show that its area is 216 sq.cm. Solution
(7) Two soldiers A and B in two different underground bunkers on a straight road, spot an intruder at the top of a hill. The angle of elevation of the intruder from A and B to the ground level in the eastern direction are 30° and 45° respectively. If A and B stand 5km apart, find the distance of the intruder from B. Solution
(8) A researcher wants to determine the width of a pond from east to west, which cannot be done by actual measurement. From a point P, he finds the distance to the eastern-most point of the pond to be 8 km, while the distance to the western most point from P to be 6 km. If the angle between the two lines of sight is 60°, find the width of the pond. Solution
(9) Two Navy helicopters A and B are flying over the Bay of Bengal at same altitude from the sea level to search a missing boat. Pilots of both the helicopters sight the boat at the same time while they are apart 10 km from each other. If the distance of the boat from A is 6 km and if the line segment AB subtends 60° at the boat, find the distance of the boat from B. Solution
(10) A straight tunnel is to be made through a mountain. A surveyor observes the two extremities A and B of the tunnel to be built from a point P in front of the mountain. If AP = 3km, BP = 5 km and ∠APB = 120^{◦}, then find the length of the tunnel to be built Solution
(11) A farmer wants to purchase a triangular shaped land with sides 120 feet and 60 feet and the angle included between these two sides is 60^{◦}. If the land costs Rs. 500 per sq.ft, find the amount he needed to purchase the land. Also find the perimeter of the land. Solution
(12) A fighter jet has to hit a small target by flying a horizontal distance. When the target is sighted, the pilot measures the angle of depression to be 30^{◦}. If after 100 km, the target has an angle of depression of 45^{◦}, how far is the target from the fighter jet at that instant? Solution
(13) A plane is 1 km from one landmark and 2 km from another. From the planes point of view the land between them subtends an angle of 45°. How far apart are the landmarks? Solution
(14) A man starts his morning walk at a point A reaches two points B and C and finally back to A such that ∠A = 60^{◦} and ∠B = 45^{◦}, AC = 4 km in the triangle ABC. Find the total distance he covered during his morning walk. Solution
(15) Two vehicles leave the same place P at the same time moving along two different roads. One vehicle moves at an average speed of 60 km/hr and the other vehicle moves at an average speed of 80 km/hr. After half an hour the vehicle reach the destinations A and B. If AB subtends 60^{◦ }at the initial point P, then find AB. Solution
(16) Suppose that a satellite in space, an earth station and the centre of earth all lie in the same plane. Let r be the radius of earth and R be the distance from the centre of earth to the satellite. Let d be the distance from the earth station to the satellite. Let 30^{◦} be the angle of elevation from the earth station to the satellite. If the line segment connecting earth station and satellite subtends angle α at the centre of earth , then prove that d = R√1 + (r/R)^{2 }− 2 (r/R) cos α. Solution
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