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How can the law of cosines be used when all three side lengths of a triangle 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.

When all three side lengths of a triangle are known (the SSS case), the law of cosines is used to calculate the measure of any of the triangle's interior angles.12

The law is expressed by the following formulas, where a, b, and c are the side lengths and A, B, and C are the angles opposite those sides, respectively:3

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

To find an unknown angle, such as angle C, you rearrange the formula to solve for the cosine of that angle:

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

By substituting the known lengths of sides a, b, and c into this equation, you can determine the value of $\cos C$ and subsequently find the angle C using the inverse cosine function. This process can be repeated for the other angles as needed to fully solve the triangle.4