koszinusztétellel

The Law of Cosines, known in Hungarian as the koszinusztétel, is a fundamental theorem in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. It is a generalization of the Pythagorean theorem to arbitrary triangles.

The theorem states that for any triangle with sides of lengths a, b, and c, and with angle C opposite side c, the following relationship holds: c² = a² + b² - 2ab cos(C). This formula can be rearranged to solve for the cosine of an angle if all three side lengths are known: cos(C) = (a² + b² - c²) / (2ab).

The Law of Cosines is particularly useful in solving triangles when two sides and the included angle are known (SAS case), or when all three sides are known (SSS case). In the SAS case, it allows for the calculation of the length of the third side. In the SSS case, it enables the determination of all three angles of the triangle.

The theorem can be derived using vector methods or by applying the Pythagorean theorem to right triangles formed by dropping an altitude within the triangle. Its applications extend beyond basic geometry to various fields of physics and engineering, such as in calculating distances and forces.