Applying the Laws of Sines and Cosines

New York State Common Core Math Geometry, Module 2, Lesson 33

Worksheets for Geometry

Student Outcomes

• Students understand that the Law of Sines can be used to find missing side lengths in a triangle when you know the measures of the angles and one side length.
• Students understand that the Law of Cosines can be used to find a missing side length in a triangle when you know the angle opposite the side and the other two side lengths.
• Students solve triangle problems using the Laws of Sines and Cosines.

Applying the Laws of Sines and Cosines

Classwork

Opening Exercise

For each triangle shown below, identify the method (Pythagorean theorem, law of sines, law of cosines) you would use to find each length 𝑥.

• Show Video

Example 1

Find the missing side length in △ 𝐴𝐵𝐶.

Example 2

Find the missing side length in △ 𝐴𝐵𝐶.

Exercises 1–6

Use the laws of sines and cosines to find all missing side lengths for each of the triangles in the exercises below. Round your answers to the tenths place.

1. Use the triangle to the right to complete this exercise.
   a. Identify the method (Pythagorean theorem, law of sines, law of cosines) you would use to find each of the missing lengths of the triangle.
   Explain why the other methods cannot be used.
   b. Find the lengths of 𝐴𝐶 and 𝐴𝐵.
2. Your school is challenging classes to compete in a triathlon. The race begins with a swim along the shore and then continues with a bike ride for 4 miles. School officials want the race to end at the place it began, so after the 4-mile bike ride, racers must turn 30° and run 3.5 miles directly back to the starting point. What is the total length of the race? Round your answer to the tenths place.
   a. Identify the method (Pythagorean theorem, law of sines, law of cosines) you would use to find the total length of the race.
   Explain why the other methods cannot be used.
   b. Determine the total length of the race. Round your answer to the tenths place.
3. Two lighthouses are 30 miles apart on each side of shorelines running north and south, as shown. Each lighthouse keeper spots a boat in the distance. One lighthouse keeper notes the location of the boat as 40° east of south, and the other lighthouse keeper marks the boat as 32° west of south. What is the distance from the boat to each of the lighthouses at the time it was spotted? Round your answers to the nearest mile.
4. A pendulum 18 in. in length swings 72° from right to left. What is the difference between the highest and lowest point of the pendulum? Round your answer to the hundredths place, and explain how you found it.
5. What appears to be the minimum amount of information about a triangle that must be given in order to use the law of sines to find an unknown length?
6. What appears to be the minimum amount of information about a triangle that must be given in order to use the law of cosines to find an unknown length?

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.