law of cosines
Technical/Very LowFormal/Academic
Definition
Meaning
A mathematical theorem relating the lengths of the sides of a triangle to the cosine of one of its angles.
A formula used in trigonometry: c² = a² + b² - 2ab cos(C). It generalises the Pythagorean theorem to any triangle, not just right-angled ones, and is used to calculate unknown sides or angles.
Linguistics
Semantic Notes
Used exclusively in mathematics, geometry, trigonometry, and related scientific/engineering fields. The term is countable, typically used with the definite article 'the', and can be preceded by words like 'apply', 'use', 'prove', or 'remember'. It's a fixed multi-word noun phrase.
Dialectal Variation
British vs American Usage
Differences
No lexical or spelling differences. Pronunciation may differ slightly due to accent.
Connotations
None beyond the mathematical meaning.
Frequency
Equal frequency in educational and technical contexts in both regions.
Vocabulary
Collocations
Grammar
Valency Patterns
use [the law of cosines] to [calculate the side]apply [the law of cosines] in [a situation]state [the law of cosines] as [c² = a² + b² - 2ab cos(C)]Vocabulary
Synonyms
Strong
Neutral
Weak
Usage
Context Usage
Business
Virtually never used.
Academic
Used in mathematics, physics, engineering textbooks, courses, and exams.
Everyday
Extremely rare outside of a specific educational context.
Technical
Common in trigonometry, navigation, computer graphics, and physics calculations involving vectors or non-right triangles.
Examples
By CEFR Level
- To find the missing side of the triangle, you need the law of cosines.
- The law of cosines is essential for solving problems in oblique triangles where the sine rule is insufficient.
- By applying the law of cosines to the vector magnitudes, the resultant force was calculated with precision.
Learning
Memory Aids
Mnemonic
Remember the phrase: 'All Can't Sin? Try Cos!' (for formula: a² = b² + c² - 2bc cos A). It connects to the sine rule ('All Can Sin' for a/sinA = b/sinB = c/sinC).
Conceptual Metaphor
A GENERALISED RELATIONSHIP (The law of cosines is a 'generalised' or 'extended' version of the simpler Pythagorean theorem.)
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Do not translate 'law' as 'закон' in a legal sense; here it means 'правило' or 'теорема' (theorem).
- Avoid direct word-for-word translation ('закон косинусов' is correct, but the concept must be understood as a specific formula).
Common Mistakes
- Confusing it with the law of sines.
- Forgetting the '-2ab cos C' part and incorrectly using the Pythagorean theorem.
- Misplacing the angle in the formula (cos C must correspond to side c).
- Omitting the definite article 'the' ('Use law of cosines' is incorrect; must be 'Use the law of cosines').
Practice
Quiz
When would you primarily use the law of cosines?
FAQ
Frequently Asked Questions
The Pythagorean theorem (a² + b² = c²) applies only to right-angled triangles. The law of cosines (c² = a² + b² - 2ab cos C) applies to all triangles and reduces to the Pythagorean theorem when angle C is 90° (since cos 90° = 0).
Use the law of cosines when you know: 1) Two sides and the included angle (SAS), or 2) All three sides (SSS) to find an angle. Use the law of sines when you know two angles and a side (AAS or ASA) or two sides and a non-included angle (SSA, the ambiguous case).
Yes. The formula can be rearranged to solve for an angle: cos(C) = (a² + b² - c²) / (2ab). This is used when you know the lengths of all three sides (SSS).
Yes, 'cosine rule' is a common synonym, particularly in British English educational contexts. 'Law of cosines' is slightly more formal and more common in American English.