Law of Cosines Calculator
What is the Law of Cosines?
The Law of Cosines is a fundamental formula in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. It generalizes the Pythagorean theorem, which applies only to right triangles.
For a triangle with sides a, b, and c, and the angle C opposite to side c, the law of cosines states:
c² = a² + b² - 2ab·cos(C)
Similarly, you can find the other angles using these formulas:
cos(A) = (b² + c² - a²) / (2bc)
cos(B) = (a² + c² - b²) / (2ac)
cos(C) = (a² + b² - c²) / (2ab)
When to Use the Law of Cosines
The Law of Cosines is particularly useful in the following scenarios:
- When you know two sides and the included angle (SAS) and need to find the third side
- When you know all three sides (SSS) and need to find an angle
- When the Law of Sines cannot be applied due to the ambiguous case
- For any triangle (not just right triangles)
Applications of the Law of Cosines
The Law of Cosines has numerous practical applications:
- Navigation and GPS systems
- Surveying and land measurement
- Architecture and construction
- Physics problems involving vectors
- Astronomy for calculating distances between celestial bodies