Cosine law

Cosine Law, also known as the Law of Cosines, is a fundamental theorem in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. This law is crucial in solving various geometric problems, especially those involving non-right triangles. The Law of Cosines generalizes the Pythagorean theorem to all types of triangles.

Statement of the Law

The Law of Cosines states that for any triangle with sides of lengths a, b, and c, and the angle opposite to side c being γ (gamma), the following equation holds:

c^2 = a^2 + b^2 - 2ab\cos(\gamma)

This equation shows that the square of the length of one side of a triangle is equal to the sum of the squares of the lengths of the other two sides, minus twice the product of those two sides and the cosine of the included angle.

Applications

The Law of Cosines is used in various fields such as mathematics, physics, and engineering for solving problems that involve calculating the dimensions of a triangle when certain other dimensions and angles are known. It is particularly useful in:

• Determining the unknown side of a triangle when two sides and the included angle are known.
• Calculating the angles of a triangle when all three sides are known.
• Solving real-world problems involving distances, such as navigation and surveying.

Derivation

The Law of Cosines can be derived using the Euclidean geometry of a triangle or by vector analysis. One common geometric derivation involves dropping a perpendicular from one vertex of the triangle to the opposite side (or its extension), dividing the triangle into two right triangles, and then applying the Pythagorean theorem and trigonometric identities.

Special Cases

• When the angle γ is a right angle (90 degrees), the Law of Cosines simplifies to the Pythagorean theorem: c^2 = a^2 + b^2, since cos(90°) = 0.
• When the triangle is equilateral, all angles are 60 degrees, and the Law of Cosines can be used to find the relationship between the side length and the angles.

See Also

• Trigonometric functions
• Sine Law
• Triangle
• Pythagorean theorem