Law of Cosines
Generalization of Pythagorean theorem
About Law of Cosines
The Law of Cosines represents generalization of Pythagorean theorem. This trigonometry formula is fundamental to mathematical analysis and serves as a cornerstone concept that students and professionals encounter throughout their mathematical journey. Its importance extends beyond pure mathematics into applied fields where quantitative analysis is required.
This formula is essential in Trigonometry and Geometry. It serves as a building block for more advanced mathematical theory and provides the foundation needed to understand complex mathematical relationships. Whether you're studying mathematics, physics, engineering, or economics, familiarity with this formula enhances your analytical capabilities.
Practical applications of the Law of Cosines include Surveying, Navigation, Physics, among others. Understanding and correctly applying this formula enables problem-solvers to approach challenges more systematically and efficiently. Mastery of this concept not only expands your mathematical knowledge but also improves your overall quantitative reasoning skills.
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LaTeX Code
c^2 = a^2 + b^2 - 2ab\cos C
Formula Information
Difficulty Level
Prerequisites
Discovered
Ancient
Discoverer
Ancient mathematicians
Real-World Applications
Examples
Mathematical Fields
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Related Topics
Important Notes
Generalizes Pythagorean theorem for any triangle. Reduces to a² + b² = c² for right triangles.
Alternative Names
Common Usage
Formula Variations
Frequently Asked Questions
What is the Law of Cosines?
The Law of Cosines generalizes the Pythagorean theorem to any triangle: c² = a² + b² - 2ab·cos(C). It relates the lengths of the three sides to one angle. For a right triangle (C = 90°), cos(90°) = 0, so it reduces to c² = a² + b² (Pythagorean theorem).
When should I use the Law of Cosines?
Use the Law of Cosines when you know: 1) Three sides (SSS) - to find angles, 2) Two sides and the included angle (SAS) - to find the third side. It's the go-to method when you don't have the angle-side combinations needed for Law of Sines.
How do I find an angle using Law of Cosines?
Rearrange the formula: cos(C) = (a² + b² - c²)/(2ab). Then C = arccos[(a² + b² - c²)/(2ab)]. For example, if sides are 3, 4, 5, then cos(C) = (9+16-25)/(2·3·4) = 0, so C = 90°. This works for any angle in the triangle.
What's the relationship to the Pythagorean theorem?
The Law of Cosines is a generalization. For a right triangle with angle C = 90°, cos(90°) = 0, so c² = a² + b² - 2ab·0 = a² + b², which is the Pythagorean theorem. The -2ab·cos(C) term accounts for non-right angles.
What are practical applications?
The Law of Cosines is used in: navigation (finding distances between points), surveying (triangulation), physics (vector addition, force resolution), engineering (structural analysis), computer graphics (distance calculations, collision detection), and any situation involving non-right triangles.
How do I choose between Law of Sines and Law of Cosines?
Use Law of Cosines for: SSS (three sides) or SAS (two sides and included angle). Use Law of Sines for: AAS (two angles and a side), ASA (angle-side-angle), or SSA (two sides and non-included angle - ambiguous case). The key is what information you're given.
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Quick Details
- Category
- Trigonometry
- Difficulty
- Intermediate
- Discovered
- Ancient
- Discoverer
- Ancient mathematicians
- Formula ID
- cosine-law