cosinustétel

The Law of Cosines, or cosinustétel in Hungarian, 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, which applies only to right-angled triangles.

For any triangle with sides of lengths a, b, and c, and with angle C opposite side c, the Law of Cosines states that c^2 = a^2 + b^2 - 2ab * cos(C). This formula can be rearranged to find any side length if the other two sides and the included angle are known, or to find an angle if all three side lengths are known.

The theorem is derived using vector algebra or by applying the Pythagorean theorem to a right-angled triangle formed by dropping an altitude from one vertex to the opposite side. The Law of Cosines has numerous applications in geometry, surveying, navigation, and physics, particularly in problems involving triangles that are not right-angled. It is a powerful tool for solving various trigonometric problems and understanding the relationships between the sides and angles of any triangle.