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How does the law of cosines help solve triangles?

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Map of the Greco-Roman world highlighting the birthplaces of notable ancient mathematicians, marked with the pi symbol and labeled with their names. Cities such as Athens, Alexandria, Syracuse, and Miletus are shown around the Mediterranean, with modern country boundaries outlined. An inset provides detail for the region of Greece and western Anatolia. — geometry This map spans a millennium of prominent Greco-Roman mathematicians, from Thales of Miletus (c. 600 bce) to Hypatia of Alexandria (c. 400 ce).

The law of cosines, also known as the cosine rule, is a generalization of the Pythagorean theorem that relates the lengths of the sides of any triangle. It is expressed by the formula:

$c^{2} = a^{2} + b^{2} - 2 a b \cos C$

In this equation, a, b, and c represent the lengths of the sides, and C is the angle opposite side c. This law is a powerful tool for solving triangles when standard right-triangle trigonometry is insufficient.12

It is primarily used to solve for unknown components in two specific scenarios:

Side-Angle-Side (SAS): When you know the lengths of two sides and the measure of the angle between them, you can use the formula to calculate the length of the third side.12
Side-Side-Side (SSS): When you know the lengths of all three sides, you can rearrange the formula to solve for any of the triangle's interior angles.12

By providing a consistent relationship between sides and angles, the law of cosines allows for the precise calculation of dimensions in any plane triangle.2