Law of Cosines Calculator

The Law of Cosines Calculator solves triangles when you know two sides and the included angle (SAS configuration). The Law of Cosines is one of the most versatile tools in trigonometry, generalizing the Pythagorean theorem to work with any triangle, not just right triangles. The fundamental formula is $$c^2 = a^2 + b^2 - 2ab\cos(C)$$, where $$C$$ is the angle between sides $$a$$ and $$b$$, and $$c$$ is the side opposite angle $$C$$.

This calculator takes sides $$a$$ and $$b$$ along with their included angle $$C$$, computes the opposite side $$c$$, then uses the Law of Cosines again to find the remaining two angles. The complete solution includes all three sides, all three angles, the area, and the perimeter. The area is computed using $$A = \frac{1}{2}ab\sin(C)$$, which follows naturally from the SAS input configuration.

The Law of Cosines is essential when the Law of Sines cannot be directly applied, particularly in SAS and SSS configurations. In the SAS case (this calculator), you know two sides and the angle between them. In the SSS case, you know all three sides and need to find the angles. Both situations require the Law of Cosines as the primary solving tool.

Practical applications are abundant. In navigation, the law determines the distance between two points when you know the distances from a reference point and the angle between the lines of sight. In physics, it calculates the resultant of two force vectors with a known angle between them. Surveyors use it when they can measure two distances and the angle between them but cannot directly measure the third distance.

The formula reveals elegant connections to the Pythagorean theorem. When $$C = 90°$$, $$\cos(90°) = 0$$, and the formula reduces to $$c^2 = a^2 + b^2$$. When $$C < 90°$$ (acute), the cosine term is positive, making $$c^2 < a^2 + b^2$$. When $$C > 90°$$ (obtuse), the cosine term is negative, making $$c^2 > a^2 + b^2$$. This provides a direct way to classify triangles by their angles.

Historically, the law was known in various forms to ancient mathematicians. Euclid proved special cases in the Elements (Propositions 12 and 13 of Book II). The general trigonometric form was developed by Al-Kashi in the 15th century, which is why the law is sometimes called Al-Kashi's theorem in some traditions.