# LAW OF SINE AND COSINE WORD PROBLEMS WORKSHEET

(1) Determine whether the following measurements produce one triangle, two triangles or no triangle:

∠B = 88°, a = 23, b = 2. Solve if solution exists.

Solution

(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 = b2 + c2 − a2/4 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 = R1 + (r/R)− 2 (r/R) cos α.              Solution

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