How is the law of cosines related to the Pythagorean theorem?

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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 generalization of the Pythagorean theorem that applies to any triangle, not just right triangles.12

The law is expressed by the formula:
$c^{2} = a^{2} + b^{2} - 2 a b \cos C$
where a, b, and c are the lengths of the sides of a triangle, and C is the angle opposite side c.12

The relationship to the Pythagorean theorem is evident when the angle C is 90 degrees. Because the cosine of 90 degrees is 0, the term $- 2 a b \cos C$ becomes zero, reducing the formula to the familiar Pythagorean theorem: $c^{2} = a^{2} + b^{2}$ . Thus, the law of cosines extends the principles of right-triangle geometry to all triangles by accounting for the angle between the sides.132