What is the Law of Cosines?

The Law of Cosines (also known as the Cosine Rule) is a fundamental theorem in trigonometry that generalizes the Pythagorean theorem to any triangle. While the Pythagorean theorem only works for right-angled triangles, the Law of Cosines allows you to find the length of a third side when two sides and the included angle are known, or to find the angles of a triangle if all three side lengths are known.

The Law of Cosines Formula

For a triangle with sides a, b, and c, and opposite angles A, B, and C, the formula is stated as:

c² = a² + b² - 2ab · cos(C)

How to Use This Calculator

This calculator is designed to help students and engineers solve triangle dimensions quickly. To use it, follow these steps:

• Side-Angle-Side (SAS): Enter side a, side b, and angle C to find the missing side c.
• Side-Side-Side (SSS): Enter all three sides (a, b, and c) to calculate the measure of angle C.
• The tool will automatically display the step-by-step mathematical derivation so you can verify your homework or project work.

Frequently Asked Questions

Q: Can the Law of Cosines be used for right triangles?
A: Yes! If angle C is 90°, cos(90°) is 0, and the formula simplifies to c² = a² + b², which is the Pythagorean theorem.

Q: What is the SAS case?
A: The SAS case occurs when you know two sides and the angle between them. This is the primary scenario where the Law of Cosines is used to find the third side.

Q: Why does the calculator require at least one side?
A: Knowing only angles (AAA) allows you to determine the shape of the triangle but not its size, resulting in infinite possible triangles.