The Law of Cosines
Formula and examples of law of cosines
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
The two common problems types that are suitable for the law of cosines are.
Problem type 1: when you know the lengths of 2 sides of a triangle and the angle in between the two sides
Problem type 2: when you know the lengths of three sides of a triangle and want to know a particular angle
Example of Law of Cosines Formula
x² = 10² + 14² −2(10)(14)cos(44°)
x² = 296 −280cos(44)
x² = 94.6
Practice applying Law of Cosines
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 X
Practice applying Law of Cosines
law of cosines to find an angle
Practice Problem 3)
Use the law of cosines to calculate x.
25² = 32² + 37² −2(32)(37)cos(x)
625 = 2393 − 2,368 cos(x)
Practice Problem 4)
 Use the law of cosines to calculate the measure of the shaded anlge.
14² = 20 ² + 12² −2(12)(20)cos(angle)
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.
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!)
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1)
Use the law of cosines formula to calculate the length of side A.
