LAW OF SINE AND COSINE WORD PROBLEMS WORKSHEET

Problem 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

Problem 2 :

If the sides of a triangle ABC are a = 4, b = 6 and c = 8, then show that 4 cosB + 3cosC = 2.            Solution

Problem 3 :

In a triangle ABC, if a = √3 − 1, b = √3 + 1 and C = 60°, find the other side and other two angles      Solution

Problem 4 :

Approaching from the west, a group of hikers records the angle of elevation to the summit of a steep mountain to be 35° at a distance of 1250 meters from the base of the  mountain. Arriving at the base of the mountain, the hikers estimate that this side of the mountain has an average slope of 48°. 

a. Find the slant height of the mountain’s west side.

b. Find the slant height of the east side of the mountain, if the east side has an average slope of 65°.

c. How tall is the mountain?

Solution

Problem 5 :

Two planes leave an airport at the same time and fly for two hours. Plane 𝐴𝐴 flies in the direction of 165° at 385 km/h and plane 𝐴𝐴 flies in the direction of 250° at 410 km/h. How far apart are the planes after two hours?

Solution

Problem 1 :

In any triangle ABC, prove that the area triangle = b² + c² − a²/4 cotA.      Solution

Problem 2 :

In a triangle ABC, if a = 12 cm, b = 8 cm and C = 30°, then show that its area is 24 sq.cm.       Solution

Problem 3 :

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

Problem 4 :

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

Problem 5 :

A stationary surveillance camera is set up to monitor activity in the parking lot of a shopping mall. If the  camera has a 38° field of vision, how many square feet of the parking lot can it tape using the given dimensions?

Solution

Problem 6 :

A New York City developer wants to build condominiums on the triangular lot formed by Greenwich, Watts, and Canal Streets. How many square meters does the  developer have to work with if the frontage along each street is approximately 34.1 m, 43.5 m, and 62.4 m, respectively?

Solution

Problem 7 :

To find the distance AB across a river, a surveyor laid off a distance BC = 354 meters on one side of the river, as shown in the accompanying figure. It is found that ∠B = 112°10′ and ∠C = 15°20′. Find the distance AB.

Solution

Problem 8 :

To determine the distance RS across a deep canyon (see
the accompanying figure), Peter lays off a distance 𝑇𝑅 = 480 meters. Then he finds that ∠𝑇 = 32° and ∠𝑅 = 102°.
Find the distance RS

Solution

Problem 1 :

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

Problem 2 :

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

Problem 3 :

The course of a boat race starts at point A in the figure and proceeds in the direction S 52^◦W to point B, then the directions S 40^◦ E to point C, and finally back to A point C lies 8 kilometers directly south of point A. Approximate the total distance of the race course.

Solution

Problem 4 :

Because of prevailing winds, a tree grew so that it was leaning 4 degree from the vertical. At the point 40 meters from the tree, the angle of elevation to the top of the tree is 30 degree. Find the height of the tree.

Solution

Problem 5 :

A plane flies 500 kilometers with bearing of 316°  from Naples to Elgin. The plane then flies 720 kilometers from Elgin to Canton (Canton is due west to Naples). Find the bearing of the flight from Elgin to Canton.

Solution

Problem 6 :

A bridge is to be built across a small lake from gazebo to a dock shown in the figure. The bearing from gazebo to the dock is S41° W. Form a tree 100 meters from gazebo, the bearings to the gazebo and dock are S74°E and S28°E respectively. Find the distance from gazebo to the dock.

Solution

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