Law of Cosines Calculator
Calculate triangle sides and angles using the Law of Cosines with step-by-step solutions
📐 Law of Cosines Calculator
Find side c using: a = 5, b = 7, ∠C = 60°
Formula
c² = a² + b² - 2ab cos(C)
Where:
- a, b, c =Triangle sides
- A, B, C =Angles opposite to sides a, b, c
- SAS =Side-Angle-Side case
- SSS =Side-Side-Side case
- cos⁻¹ =Inverse cosine function
About Law of Cosines
The Law of Cosines relates the lengths of a triangle's sides to the cosine of one of its angles. It's a generalization of the Pythagorean theorem.
When to Use:
- • SAS: Two sides and the included angle known
- • SSS: All three sides known, find angles
- • When Law of Sines gives ambiguous results
- • For obtuse triangles
Alternative Forms:
- • a² = b² + c² - 2bc cos(A)
- • b² = a² + c² - 2ac cos(B)
- • c² = a² + b² - 2ab cos(C)
Applications:
- • Navigation and GPS systems
- • Engineering and construction
- • Astronomy and satellite tracking
- • Computer graphics and game development
- • Surveying and land measurement
Special Cases:
- • When C = 90°: c² = a² + b² (Pythagorean theorem)
- • When C = 60°: c² = a² + b² - ab
- • When C = 120°: c² = a² + b² + ab