MathIsimple
Cosine Law Calculator

Cosine Law Calculator

Solve triangles using the Law of Cosines. Calculate unknown sides and angles with automatic triangle type classification and step-by-step solutions.

Geometry Spot/Calculators/Cosine Law

Cosine Law Calculator

Solve triangles using the cosine law. Classify triangle types and get step-by-step solutions.

Learn Cosine Law

Cosine Law Calculator

Enter known sides and angles to solve the triangle

Current Setup:
Triangle ABC: a=?, b=?, c=?, A=?°, B=?°, C=?°
Side a is opposite angle A, side b is opposite angle B, side c is opposite angle C

Cosine Law Formulas

a2=b2+c2−2bccos⁡Aa^2 = b^2 + c^2 - 2bc\cos A
b2=a2+c2−2accos⁡Bb^2 = a^2 + c^2 - 2ac\cos B
c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C
When to use:
  • SAS: Two sides and the included angle
  • SSS: All three sides are known
  • Finding angles when all sides are known
Triangle Types: Acute, Right, or Obtuse based on side relationships

Related Resources

Cosine Law Applications

Key Points:
  • • SAS and SSS cases have unique solutions
  • • Generalizes Pythagorean theorem to all triangles
  • • Can classify triangle type from side lengths
  • • When C = 90°, reduces to a² + b² = c²

Calculator Features

SAS Case

Two sides and included angle - find the third side

c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C

SSS Case

All three sides known - find all angles

cos⁡C=a2+b2−c22ab\cos C = \frac{a^2 + b^2 - c^2}{2ab}

Triangle Classification

Automatically classify as acute, right, or obtuse

c2≶a2+b2c^2 \lessgtr a^2 + b^2

The Cosine Law Formula

Side Form

c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C

Angle Form

cos⁡C=a2+b2−c22ab\cos C = \frac{a^2 + b^2 - c^2}{2ab}

When to Use Cosine Law

  • ✓SAS: Two sides and the included angle
  • ✓SSS: All three sides are known
  • iCosine law always gives a unique solution (no ambiguous cases)

Triangle Classification Using Cosine Law

Acute Triangle

All angles less than 90°

c2<a2+b2c^2 < a^2 + b^2

Right Triangle

One angle exactly 90°

c2=a2+b2c^2 = a^2 + b^2

Obtuse Triangle

One angle greater than 90°

c2>a2+b2c^2 > a^2 + b^2

Related Tools & Learning

Cosine Law Lesson

Learn the theory and applications of cosine law

Sine Law Calculator

For AAS and ASA triangle problems

Triangle Area Calculator

Calculate triangle area using multiple methods

Frequently Asked Questions

The Law of Cosines states that c² = a² + b² - 2ab·cos(C), where a, b, c are sides of a triangle and C is the angle opposite to side c. It's a generalization of the Pythagorean theorem for any triangle.
Ask AI ✨