law of sines
LowTechnical/Formal
Definition
Meaning
A theorem in trigonometry stating that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant and equal to the diameter of the triangle's circumcircle. For triangle ABC, it is expressed as: a/sin A = b/sin B = c/sin C.
A fundamental trigonometric rule used to solve for unknown angles and sides in non-right (oblique) triangles, particularly in applications of trigonometry to surveying, navigation, physics, and engineering. It is foundational for understanding triangle relationships.
Linguistics
Semantic Notes
This is a fixed technical term. It is always used in its full form 'law of sines' and is never shortened. It belongs to the semantic field of mathematics/trigonometry. The term is singular ('law'), not plural.
Dialectal Variation
British vs American Usage
Differences
No significant lexical or semantic differences. The term is identical in both varieties. However, 'sine rule' is a common, equivalent alternative in British English educational contexts, while 'law of sines' is standard in American textbooks.
Connotations
Identical technical connotations. 'Law of sines' carries a slightly more formal, theorem-like weight, whereas 'sine rule' can sound more pedagogical.
Frequency
In British educational materials, 'sine rule' is more frequent, while in American and international academic texts, 'law of sines' is dominant.
Vocabulary
Collocations
Grammar
Valency Patterns
[Subject: One/We/This method] + [Verb: uses/applies/employs/involves] + the law of sines + [to-infinitive: to solve/find/calculate] + [Object: a side/an angle/the area].The law of sines + [Verb: holds/is valid/states] + [that-clause].Vocabulary
Synonyms
Strong
Neutral
Weak
Vocabulary
Antonyms
Phrases
Idioms & Phrases
- “There are no idioms containing this term.”
Usage
Context Usage
Business
Virtually never used.
Academic
Used exclusively in mathematics, physics, engineering, and related STEM fields in textbooks, lectures, and problem sets.
Everyday
Extremely rare. Only used when discussing specific math homework or technical tasks.
Technical
Core term in trigonometry, computational geometry, surveying, and related technical disciplines.
Examples
By Part of Speech
verb
British English
- We need to sine-rule this triangle to find the missing side.
American English
- We need to apply the law of sines to this triangle.
adverb
British English
- The side was calculated sine-rule-wise.
American English
- The problem was solved using the law of sines.
adjective
British English
- The sine-rule method is appropriate here.
American English
- The law-of-sines approach is the most efficient.
Examples
By CEFR Level
- This is too difficult for A2 level.
- In maths class, we learned a new rule called the law of sines.
- To find the unknown angle in the oblique triangle, you must apply the law of sines correctly.
- The surveyor employed the law of sines to triangulate the position of the distant landmark, compensating for the inaccessible terrain.
Learning
Memory Aids
Mnemonic
Sides and Sines Stay Symmetric. (The three ratios of side to sine of its opposite angle are all equal.)
Conceptual Metaphor
BALANCE / PROPORTIONALITY. The law conceptualises a triangle as a balanced system where sides and their opposing angles are in a fixed proportional relationship.
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Do not translate as 'синусоидальный закон', which refers to sinusoidal law in physics/wave theory. The correct translation is 'теорема синусов'.
- Avoid literal translation of 'law' as 'закон' in a legal sense; here it is a mathematical 'law' or 'theorem' ('теорема').
Common Mistakes
- Misapplying it to right-angled triangles when simpler methods exist.
- Using it for the Side-Side-Side (SSS) or Side-Angle-Side (SAS) cases where the law of cosines is required.
- Incorrectly setting up the proportion (e.g., sin(angle)/side instead of side/sin(angle)).
- Forgetting that it can yield an ambiguous case (SSA) with two possible solutions.
Practice
Quiz
The law of sines is primarily used to solve which type of triangle?
FAQ
Frequently Asked Questions
Use the law of sines when you know either: 1) Two angles and any side (AAS or ASA), or 2) Two sides and an angle opposite one of them (SSA – the ambiguous case). Use the law of cosines for SAS or SSS configurations.
Yes, it works for all triangles, right and oblique. However, for right triangles, using basic trigonometric ratios (SOH CAH TOA) is often simpler.
The ambiguous case (SSA) occurs when you know two sides and a non-included angle. Depending on the values, there may be zero, one, or two possible triangles that satisfy the given information.
Yes, they are synonyms. 'Sine rule' is more common in British English educational contexts, while 'law of sines' is the standard term in American English and formal international mathematics.