Worksheet: Applications of the Law of Cosine

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

Q1:

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.

Q2:

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

• A 52.9 miles,
• B 172.9 miles,
• C miles,
• D 172.9 miles,
• E 362.9 miles,

Q3:

A 113-foot tower is located on a hill that is inclined 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.

Q4:

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

Q5:

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

Q6:

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 , 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.

Q7:

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

Q8:

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