# Law of Cosines

Definition: The law of cosines is used to find a triangle’s unknown sides or angles. It relates the side lengths of a triangle to the cosine of one of its angles with the following equations:

The law of cosines can only be used when the following information is available:

Two side lengths and the angle between them (the included angle) are known

OR

Three side lengths are known

Example: Find the missing angles:Solution:

Step 1: Given side a = 5, side b = 6 and side c = 4, set up the equation to find ∠C:

Step 2: Simplify further to find ∠C:

Step 3: Now that we know one side length and its opposite angle plus two other side lengths, we can use the law of sines to complete the triangle. Plug all available information into the formula:

Step 4: Working with sides b and c, we can find ∠B:

Step 5: To find the measure of ∠A, we can use the Triangle Angle Sum Theorem (the sum of a triangle’s interior angles always totals 180°):

∠A = 180 – 41.4 – 82.8 = 55.8°

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