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How can the law of cosines find a missing side when two sides and their included angle are known?

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Diagram of a right triangle labeled with sides a, b, and hypotenuse c, alongside formulas for the six trigonometric functions-sine, cosine, tangent, cosecant, secant, and cotangent-each defined by ratios of the triangle's sides. — trigonometry Based on the definitions, various simple relationships exist among the functions. For example, csc A = 1/sin A, sec A = 1/cos A, cot A = 1/tan A, and tan A = sin A/cos A.

The law of cosines is a powerful tool in trigonometry used to determine the length of an unknown side of a triangle when two sides and their included angle are known (a condition known as SAS).

To find the missing side, you use the following formula:

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

In this equation:
* $a$ and $b$ represent the lengths of the two known sides.
* $C$ is the measure of the angle located between sides $a$ and $b$ .
* $c$ is the length of the side opposite to angle $C$ , which is the side you are solving for.

By substituting the known values into the equation and calculating the result, you can find $c^{2}$ and then take the square root to determine the length of the missing side. This formula serves as a generalization of the Pythagorean theorem, extending its utility beyond right triangles to any plane triangle.123