Worksheet: Applications of the Cosine Law

In this worksheet, we will practice using the cosine law to solve real-world problems.

Q1:

A person rode a bicycle 7 2 km east, and then he rode for another 2 1 4 5 k m south of east. Determine the magnitude and direction of the displacement, rounding the direction to the nearest minute.

  • A 7 1 7 km, 5 9 2 south of east
  • B 7 1 1 km, 6 4 4 6 south of east
  • C 7 1 1 km, 2 5 1 4 south of east
  • D 7 1 7 km, 3 0 5 8 south of east

Q2:

𝐴 𝐵 𝐶 𝐷 is a trapezium, where 𝐴 𝐷 / / 𝐵 𝐶 , 𝐴 𝐷 = 1 9 c m , 𝐴 𝐵 = 2 0 c m , 𝐵 𝐶 = 1 1 c m , and 𝑚 𝐴 = 1 1 5 . Find the lengths of 𝐵 𝐷 and 𝐶 𝐷 , giving the answer to one decimal place.

  • A 𝐵 𝐷 = 2 0 . 4 c m , 𝐶 𝐷 = 1 7 . 9 c m
  • B 𝐵 𝐷 = 3 0 . 4 c m , 𝐶 𝐷 = 2 3 . 1 c m
  • C 𝐵 𝐷 = 3 2 . 9 c m , 𝐶 𝐷 = 3 0 . 0 c m
  • D 𝐵 𝐷 = 3 2 . 9 c m , 𝐶 𝐷 = 2 4 . 5 c m
  • E 𝐵 𝐷 = 3 0 . 4 c m , 𝐶 𝐷 = 2 0 . 0 c m

Q3:

In a football match, the midfielder was 20 m away from the right-wing player. The midfielder turned 1 3 1 and saw the left-wing player 12 m away from him. Find the distance between the right-wing player and the left-wing player giving the answer to one decimal place.

Q4:

Two ships sailed from the same point. One ship sailed in a direction of 5 2 3 9 1 4 west of south at a speed of 5 km/h whereas the other sailed in a direction of 3 6 7 4 8 north of west at a speed of 18 km/h. Find the distance between the two ships after 3 hours giving the answer to two decimal places.

Q5:

Philadelphia is 140 miles from Washington, DC. Washington, DC, is 442 miles from Boston. Boston is 315 miles from Philadelphia. Find the angles in the triangle whose vertices are at Philadelphia, Washington, DC, and Boston.

  • A 3 . 8 8 , 1 1 5 . 7 1 , 6 0 . 4 1
  • B 6 2 . 1 2 , 6 0 . 4 1 , 5 7 . 4 7
  • C 2 . 1 7 , 6 2 . 1 2 , 1 1 5 . 7 1
  • D 9 . 0 6 , 2 0 . 7 4 , 1 5 0 . 2 0
  • E 6 2 . 1 2 , 2 0 . 7 4 , 9 7 . 1 4

Q6:

Los Angeles is 1744 miles from Chicago, Chicago is 712 miles from New York, and New York is 2451 miles from Los Angeles. Find the angles in the triangle with its vertices at the three cities.

  • A 4 . 6 6 , 5 . 7 3 , 1 6 9 . 6 1
  • B 2 . 3 3 , 1 9 . 4 4 , 1 5 8 . 2 3
  • C 8 4 . 2 7 , 2 . 3 3 , 9 3 . 4
  • D 2 . 3 3 , 5 . 7 3 , 1 7 1 . 9 4
  • E 2 . 3 3 , 1 1 . 6 4 , 1 6 6 . 0 3

Q7:

Two sides of a triangular flower bed measure 40 feet and 65 feet with an angle of 5 0 between them. Calculate the length of the third side, giving your answer to one decimal place.

Q8:

A surveyor takes measurements as shown in the diagram. Calculate the distance across the lake. Give your answer to one decimal place.

Q9:

A farmer wants to fence off a triangular piece of land. The lengths of two sides of the fence are 72 metres and 55 metres, and the angle between them is 8 3 . Find the perimeter of the fence giving your answer to the nearest metre.

Q10:

𝐴 𝐵 𝐶 𝐷 is a quadrilateral where 𝑚 𝐷 𝐴 𝐵 = 𝑚 𝐷 𝐵 𝐶 = 9 0 , 𝐵 𝐷 = 2 6 c m , 𝐴 𝐷 = 2 4 c m and 𝑚 𝐷 𝐶 𝐵 = 3 2 . Find the length of 𝐴 𝐶 giving the answer to the nearest centimetre.

Q11:

An engineer is standing at point 𝐴 where point 𝐵 is 1 400 metres away, point 𝐶 is 200 metres away, and 𝑚 𝐵 𝐴 𝐶 is 1 0 5 . Find the distance between points 𝐵 and 𝐶 giving the answer in kilometres to two decimal places.

Q12:

𝐴 𝐵 𝐶 𝐷 is a quadrilateral where 𝐴 𝐵 = 3 1 c m , 𝐵 𝐶 = 4 0 c m , 𝐶 𝐷 = 4 3 c m , 𝐷 𝐴 = 3 2 c m and 𝑚 𝐴 = 6 4 . Find 𝑚 𝐶 𝐵 𝐷 giving the answer to the nearest degree.

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