The Law of Cosines

Law of Cosines Formula

The law of cosines is a formula that relates the three sides of a triangle to the cosine of a given angle. This formula allows you

• Case I) to calculate the side length of non-right triangle as long as you know two sides and an angle (See picture below)
• Case 2) to calculate any angle of a triangle if you know all three side lengths

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Video Tutorial of Law on Law of Cosines

Interactive Demonstration of the Law of Cosines Formula

The interactive demonstration below illustrates the Law of cosines formula in action. Drag around the points in the triangle to observe who the formula works.

Example of Law of Cosines Formula

x² = 10² + 14² −2(10)(14)cos(44°)

x² = 296 −280cos(44)

x² = 94.6

Practice Problem 1) Use the law of cosines formula to calculate the length of side A.

Answer

A² = 20² + 13² −2(20)(13)cos(66)

A² = 400 + 169 −520cos(66)

A² = 569 −211

A² = 358

Practice Problem 2) Use the law of cosines formula to calculate the length of side A.

Step 1: set up the formula

A² = 52² + 16² −2(52)(16)cos(115°)

Step 2: do the calculation

A² = 2,704 + 256 − 1,664cos(115)

A² = 2,960− 1,664(-.423)

A² = 2,960 +703 = 3663

Practice Problem 3)Use the law of cosines to calculate the lenght of side x.

Answer

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Practice Problem 3) Use the law of cosines to calculate x.

Step 1: set up formula

25² = 32² + 37² −2(32)(37)cos(x)

Solve the equation

625 = 2393 − 2,368 cos(x)

Practice Problem 4) Use the law of cosines to calculate the measure of the shaded anlge.

Step 1: set up formula

14² = 20 ² + 12² −2(12)(20)cos(angle)

Solve the equation

Solve the equation in terms of cos(angle) and then use cos^-1 to determine the angle measurement

The law of Cosines and the pythagorean theorem.

Use the law of cosines to calculate the value of x. This exercise should help you see the connection between the law of cosines and the pythagorean theorem.

Explanation

Using the formula for the law of cosines:

x² = 21² + 73.24² −2(21)(73.24)cos90°

cos90° = 0

x² = 21² + 73.24² − 0

x² = 21² + 73.24² (this is the formulation of the pythagorean theorm for the triangle on the left!)