Law of Cosines Calculator
Side c: -
Angle C: -°
Understanding the Law of Cosines
What is the Law of Cosines?
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.
General Formula:
c² = a² + b² - 2ab·cos(C)
where:
- a, b, c are the sides of the triangle
- C is the angle opposite to side c
- cos(C) is the cosine of angle C
Applications and Uses
Surveying
Used in land surveys and distance calculations
- Property boundary measurements
- Terrain mapping
- Distance calculations
Navigation
Essential in maritime and aerial navigation
- Course plotting
- Distance estimation
- Position fixing
Engineering
Applied in various engineering fields
- Structural analysis
- Force calculations
- Vector problems
Physics
Used in vector calculations
- Force resolution
- Momentum analysis
- Wave mechanics
Special Cases and Variations
Right Triangle Case:
When C = 90°, cos(C) = 0, reducing to the Pythagorean theorem:
c² = a² + b²
Obtuse Triangle Case:
When C > 90°, cos(C) is negative, leading to:
c² = a² + b² + 2ab·|cos(C)|
Acute Triangle Case:
When C < 90°, standard formula applies:
c² = a² + b² - 2ab·cos(C)
Historical Context
The Law of Cosines generalizes the Pythagorean theorem and was first discovered by ancient mathematicians:
- Ancient Greek origins (Euclid's Elements)
- Persian mathematician Al-Kashi's contributions
- Modern formulation in trigonometry
- Extensions to spherical trigonometry