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:

𝐴𝐡𝐢 is a triangle, where 𝐷 is the midpoint of 𝐡𝐢, π‘šβˆ π΅=28∘, π‘šβˆ π΄=69∘, and π‘Ž=20cm. Find the length of 𝐴𝐷 giving the answer to two decimal places.

Q2:

𝐴𝐡𝐢 is a triangle, where π‘Ž=19 cm, 𝑏=9 cm, and π‘šβˆ πΆ=45∘. Find the radius of the circumcircle giving the answer to two decimal places.

Q3:

𝐴𝐡𝐢 is a triangle where π‘Ž=26 cm, 𝑏=22 cm and 𝑐=6 cm. Find the radius of the circumcircle giving the answer to three decimal places.

Q4:

𝐴𝐡𝐢 is a triangle where π‘šβˆ π΄=30∘, the ratio between 𝑏 and 𝑐 is √3∢2, and the area of the circumcircle is 225πœ‹ cm2. Find the perimeter of triangle 𝐴𝐡𝐢 giving the answer to the nearest centimeter.

Q5:

𝐴𝐡𝐢𝐷 is a quadrilateral where 𝐴𝐡=16cm, π‘šβˆ π΄π·π΅=40∘, π‘šβˆ π·π΅π΄=100∘, 𝐡𝐢=21cm and 𝐷𝐢=9cm. Find π‘šβˆ π΅πΆπ· giving the answer to the nearest second and the area of 𝐡𝐢𝐷 giving the answer to three decimal places.

  • A11110β€²6β€²β€²βˆ˜, 88.123 cm2
  • B2333β€²23β€²β€²βˆ˜, 37.767 cm2
  • C6923β€²58β€²β€²βˆ˜, 134.283 cm2
  • D4516β€²30β€²β€²βˆ˜, 67.142 cm2

Q6:

𝐴𝐡𝐢𝐷 is a quadrilateral where π‘šβˆ π΄π΅πΆ=90∘, π‘šβˆ π΅π΄π·=41∘, 𝐴𝐡=𝐴𝐷=30.9cm and 𝐡𝐷=𝐡𝐢. Find the area of 𝐴𝐡𝐢𝐷 giving the answer to two decimal places.

Q7:

Find the area of 𝐴𝐡𝐢𝐷 given 𝐸 is the point of intersection of 𝐴𝐢 and 𝐡𝐷, 𝐴𝐸=5cm, 𝐸𝐢=8.9cm, 𝐸𝐷=7.7cm, and π‘šβˆ π΄πΈπ΅=80∘. Give the answer to the nearest square centimeter.

Q8:

𝐴𝐡𝐢𝐷 is a parallelogram where 𝑀 is the point of intersection to the diagonals, 𝐴𝐢=21.1cm, π‘šβˆ π΄π‘€π·=8054β€²βˆ˜ and π‘šβˆ πΆπ΄π΅=5354β€²βˆ˜. Find the area of the parallelogram giving the answer to two decimal places.

Q9:

In a parallelogram 𝐴𝐡𝐢𝐷, 𝐡𝐢=8inches and 𝐴𝐡=5inches, where π‘šβˆ π΄π΅πΆ=134∘. Work out the length of 𝐴𝐢. Give your answer to 2 decimal places.

Q10:

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

Q11:

In a triangle 𝐴𝐡𝐢, 𝐴𝐡=3, 𝐴𝐢=8, and π‘šβˆ π΅π΄πΆ=52∘.

If 𝐡𝐷 is the altitude from 𝐡, what is 𝐡𝐷 to 2 decimal places?

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

Q12:

Find the area of the green part of the diagram, given that π‘šβˆ πΆπ΄π΅=77∘, π‘šβˆ π΅πΆπ΄=57∘, and 𝐢𝐡=19cm. Give the answer to the nearest square centimeter.

Q13:

𝐴𝐡𝐢 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.

Q14:

𝐴𝐡𝐢 is a triangle where the ratio between π‘šβˆ π΄, π‘šβˆ π΅, and π‘šβˆ πΆ is 5∢4∢3 and the radius of the circumcircle is 6 cm. Find the area of 𝐴𝐡𝐢 giving the answer to two decimal places.

Q15:

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?

Q16:

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.

Q17:

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.

Q18:

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.

Q19:

An airplane flies 220 miles on a bearing of 40∘ and then changes course and flies 180 miles on a bearing of 170∘. How far is the plane from its starting point and on what bearing? Give your answers to one decimal place.

  • A172.9 miles, 52.9∘
  • B172.9 miles, 92.9∘
  • C52.9 miles, 92.9∘
  • D362.9 miles, 102.5∘
  • E6,141.9 miles, 37.1∘

Q20:

Two boats leave a harbor in different directions. One travels due west at 30 mph, and the other travels on a bearing of 295∘ at 42 mph. How far apart are they after 90 minutes?

Q21:

Two ships left a port at the same time. One traveled on a bearing of 320∘ at 18 mph, while the other traveled on a bearing of 194∘ at 22 mph. Find the distance between the ships after 10 hours. Give your answer to the nearest mile.

Q22:

A plane follows a straight path at a steady speed of 600 miles per hour. After 1 hour  30 minutes, the pilot makes a course correction of 10∘, and the flight continues at the same speed, following a straight path, for another 2 hours. How far is the plane from its initial position? Give your answer to the nearest mile.

Q23:

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.

  • A9.06∘, 20.74∘, 150.20∘
  • B2.17∘, 62.12∘, 115.71∘
  • C3.88∘, 115.71∘, 60.41∘
  • D62.12∘, 60.41∘, 57.47∘
  • E62.12∘, 20.74∘, 97.14∘

Q24:

Los Angeles is 1,744 miles from Chicago, Chicago is 712 miles from New York, and New York is 2,451 miles from Los Angeles. Find the angles in the triangle with its vertices at the three cities.

  • A4.66∘, 5.73∘, 169.61∘
  • B2.33∘, 19.44∘, 158.23∘
  • C84.27∘, 2.33∘, 93.4∘
  • D2.33∘, 5.73∘, 171.94∘
  • E2.33∘, 11.64∘, 166.03∘

Q25:

Two sides of a triangular flower bed measure 40 feet and 65 feet with an angle of 50∘ between them. Calculate the length of the third side, giving your answer to one decimal place.

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