Law of Sines and Cosines

How to determine which formula to use

The goal of this page is to help students better understand when to use the law of sines and when to use the Law of Cosines

Picture of Law of Sines and Cosines

Practice Problems

Problem 1

Look at each triangle below and, based on the given information, decide whether you could use the Law of Sines, the Law of Cosines (or neither)

Law of Sines and Cosines problem

Triangle 1 and 2 are clear candidates for the law of sines

Triangle 1 (Top left) - Law of Sines
Triangle 2 (Top Left)
Triangle 3( Bottom Left)

To get the measure of x . You wold first need to use the law of cosines to get the length of the side opposite the 115° triangle. From there you could use the law of sines

Triangle 4( Bottom Right)

Since the sum of the measure of the interior angles of a triangle is 180°, you can find the measure of the 3rd angle . From there you could use the law of sines to calculate the length of side x.

Problem 2

Decide which formula (law of Sines/Cosines) you would use to calculate the value of x below? After you decide that, try to set up the equation(Do not solve--just substitute into the proper formula)

Since you know 3 sides, and are trying to find an angle this is Law of Cosines problem. 8² = 5² + 6² -2(5)(6)(cos(x))

Problem 3

Decide which formula (law of Sines/Cosines) you would use to calculate the value of x below? After you decide that, try to set up the equation(Do not solve--just substitute into the proper formula)

Since you know 2 sides , their included angle, and you are trying to find the side length opposite the angle, this is Law of Cosines problem. x² = 11² + 7² -2(11)(7)(cos(50))

Problem 4

Decide which formula (law of Sines/Cosines) you would use to calculate the value of x below? After you decide that, try to set up the equation(Do not solve--just substitute into the proper formula)

Since you know a side length (11) and its opposite angle (50) and want to the angle measurement opposite the length of side 7, this is a law of sines problem