Worksheet: Applications on Sine and Cosine Laws

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

Q1:

𝐴 𝐵 𝐶 is a triangle, where 𝐷 is the midpoint of 𝐵𝐶, 𝑚𝐵=28, 𝑚𝐴=69, and 𝑎=20cm. Find the length of 𝐴𝐷 giving the answer to two decimal places.

Q2:

𝐷 is the midpoint of 𝐵𝐶 and 𝑚𝐵𝐴𝐷=37. Find tan𝐷𝐴𝐶 giving the answer to three decimal places.

Q3:

𝐴 𝐵 𝐶 is a triangle where sinsinsin𝐴25=𝐵29=𝐶15. Find the largest angle in 𝐴𝐵𝐶 giving the answer to the nearest degree.

Q4:

𝐴 𝐵 𝐶 is a triangle where 𝑎=44cm, 𝑏=31cm and 𝑚𝐶=69. Point 𝐷 lies on 𝐴𝐵 such that 𝐶𝐷𝐴𝐷. Find the length of 𝐶𝐷 giving the answer to two decimal places.

Q5:

𝐴 𝐵 𝐶 is a triangle, where 𝑎=19 cm, 𝑏=9 cm, and 𝑚𝐶=45. Find the radius of the circumcircle giving the answer to two decimal places.

Q6:

𝐴 𝐵 𝐶 is a triangle where 𝑎=26 cm, 𝑏=22 cm and 𝑐=6 cm. Find the radius of the circumcircle giving the answer to three decimal places.

Q7:

𝐴 𝐵 𝐶 is a triangle where 𝑚𝐴=30, the ratio between 𝑏 and 𝑐 is 32, and the area of the circumcircle is 225𝜋 cm2. Find the perimeter of triangle 𝐴𝐵𝐶 giving the answer to the nearest centimeter.

Q8:

𝐴 𝐵 𝐶 𝐷 is a quadrilateral where 𝐴𝐵=16cm, 𝑚𝐴𝐷𝐵=40, 𝑚𝐷𝐵𝐴=100, 𝐵𝐶=21cm and 𝐷𝐶=9cm. Find 𝑚𝐵𝐶𝐷 giving the answer to the nearest second and the area of 𝐵𝐶𝐷 giving the answer to three decimal places.

  • A 1 1 1 1 0 6 , 88.123 cm2
  • B 2 3 3 3 2 3 , 37.767 cm2
  • C 6 9 2 3 5 8 , 134.283 cm2
  • D 4 5 1 6 3 0 , 67.142 cm2

Q9:

𝐴 𝐵 𝐶 𝐷 is a quadrilateral where 𝑚𝐴𝐵𝐶=90, 𝑚𝐵𝐴𝐷=41, 𝐴𝐵=𝐴𝐷=30.9cm and 𝐵𝐷=𝐵𝐶. Find the area of 𝐴𝐵𝐶𝐷 giving the answer to two decimal places.

Q10:

Find the area of 𝐴𝐵𝐶𝐷 given 𝐸 is the point of intersection of 𝐴𝐶 and 𝐵𝐷, 𝐴𝐸=5cm, 𝐸𝐶=8.9cm, 𝐸𝐷=7.7cm, and 𝑚𝐴𝐸𝐵=80. Give the answer to the nearest square centimeter.

Q11:

𝐴 𝐵 𝐶 𝐷 is a trapezium, where 𝐴𝐷𝐵𝐶, 𝐴𝐷=8cm, 𝑚𝐵=68, 𝑚𝐷=124, and 𝑚𝐴𝐶𝐵=30. Find the lengths of 𝐴𝐶 and 𝐵𝐶 giving the answer to the nearest centimetre.

  • A 𝐴 𝐶 = 1 5 c m , 𝐵 𝐶 = 4 c m
  • B 𝐴 𝐶 = 4 c m , 𝐵 𝐶 = 8 c m
  • C 𝐴 𝐶 = 1 6 c m , 𝐵 𝐶 = 1 5 c m
  • D 𝐴 𝐶 = 1 5 c m , 𝐵 𝐶 = 1 6 c m
  • E 𝐴 𝐶 = 4 c m , 𝐵 𝐶 = 1 6 c m

Q12:

𝐴 𝐵 𝐶 𝐷 is a trapezium where 𝐴𝐷𝐵𝐶, 𝐵𝐶=20cm, 𝑚𝐴𝐵𝐶=61, 𝐴𝐷=11cm and 𝑚𝐵𝐴𝐶=59. Find the area of the trapezium giving the answer to the nearest square centimetre.

Q13:

𝐴 𝐵 𝐶 𝐷 is a trapezium, where 𝐴𝐷𝐵𝐶, 𝐴𝐷=20cm, 𝑚𝐵=55, 𝑚𝐷=80, and 𝑚𝐴𝐶𝐵=52. Find the area of the trapezium giving the answer to the nearest square centimetre.

Q14:

𝐴 𝐵 𝐶 𝐷 is a parallelogram where 𝑀 is the point of intersection to the diagonals, 𝐴𝐶=21.1cm, 𝑚𝐴𝑀𝐷=8054 and 𝑚𝐶𝐴𝐵=5354. Find the area of the parallelogram giving the answer to two decimal places.

Q15:

𝐴 𝐵 𝐶 is a triangle where 𝐴𝐵=7cm, 𝑚𝐵=60 and the area of the triangle is 913 cm2. Find the perimeter of 𝐴𝐵𝐶 giving the answer to two decimal places.

Q16:

In a parallelogram 𝐴𝐵𝐶𝐷, 𝐵𝐶=8inches and 𝐴𝐵=5inches, where 𝑚𝐴𝐵𝐶=134. Work out the length of 𝐴𝐶. Give your answer to 2 decimal places.

Q17:

The side length of a regular pentagon 𝐴𝐵𝐶𝐷𝐸 is 25.81 cm. Find the length of the diagonal 𝐴𝐶 giving the answer to two decimal places.

Q18:

In a triangle 𝐴𝐵𝐶, 𝐴𝐵=3, 𝐴𝐶=8, and 𝑚𝐵𝐴𝐶=52.

If 𝐵𝐷 is the altitude from 𝐵, what is 𝐵𝐷 to 2 decimal places?

Calculate the area of the triangle 𝐴𝐵𝐶 to 2 decimal places.

Q19:

Find the area of the green part of the diagram, given that 𝑚𝐶𝐴𝐵=77, 𝑚𝐵𝐶𝐴=57, and 𝐶𝐵=19cm. Give the answer to the nearest square centimeter.

Q20:

𝐴 𝐵 𝐶 is an equilateral triangle inscribed in circle 𝑀 with a radius of 95 cm. Find the area of each shaded segment giving the answer to the nearest square centimeter.

Q21:

𝐴 𝐵 𝐶 is a triangle where the ratio between 𝑚𝐴, 𝑚𝐵, and 𝑚𝐶 is 543 and the radius of the circumcircle is 6 cm. Find the area of 𝐴𝐵𝐶 giving the answer to two decimal places.

Q22:

Two planes leave the same airport at the same time. One flies on a bearing of 20 at 500 miles per hour. The other flies on a bearing of 30 at 600 miles per hour. How far apart are the planes after 2 hours?

Q23:

The shown diagram represents a satellite calculating the distances and the angle. Find the distance between the two cities. Give your answer to one decimal place. Note that the diagram is not to scale.

Q24:

A 113-foot tower is located on a hill that is inclined 34 to the horizontal, as shown in the figure. A guy wire is to be attached to the top of the tower and anchored at a point 98 feet uphill from the base of the tower. Find the length of wire needed.

Q25:

A man ran 24 km down a straight road. He then turned 139 and ran a further 9 km along another straight road. Find, to the nearest kilometer, the shortest distance between his start and end points.

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