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Showing 1651–1680 of 1709 reference entries

Mathematics & logicOrder of arithmetic operations

Grouping and equal-priority arithmetic evaluation.

MethodMathematics & logicScientific notation

A normalized decimal coefficient with an integer power of ten.

DefinitionMathematics & logicDirect and inverse proportion

Constant-ratio and constant-product relationships.

DefinitionMathematics & logicAngles, intersecting lines and parallel lines

Euclidean line-angle relationships.

Mathematical statementMathematics & logicAngle sums of polygons

Interior and exterior angle sums for convex polygons.

Mathematical statementMathematics & logicSine and cosine rules for triangles

Side-angle laws beyond right triangles.

Mathematical statementMathematics & logicPigeonhole principle

A finite assignment forces a shared output when inputs outnumber outputs.

Mathematical statementMathematics & logicTotal probability and event Bayes' theorem

Finite-partition probabilities and reversal of event conditioning.

Mathematical statementMathematics & logicLeast common multiple

The smallest positive common multiple of positive integers.

DefinitionMathematics & logicPolynomials and degree

Finite sums of nonnegative integer powers with fixed coefficients.

DefinitionMathematics & logicDecimal rounding and significant digits

Decimal places, retained significant digits and rounding ties.

MethodMathematics & logicCone and sphere volume and surface area

Euclidean cone and sphere mensuration.

Mathematical statementMathematics & logicCircle angles and tangents

Inscribed angles and radius-tangent perpendicularity.

Mathematical statementMathematics & logicStraight-line equations

Slope-intercept form of a nonvertical line.

Mathematical statementInformationProgram names, bindings and assignment

Names refer to values through an execution environment.

DefinitionInformationSequence, conditionals and loops

Selection and repetition determine which program statements execute.

DefinitionInformationFunction definitions, parameters and calls

A call connects arguments with local parameter bindings.

DefinitionInformationRecursive procedures and base cases

Self-calls reduce a problem toward a terminating case.

MethodInformationData types and checking

Types describe values and permitted operations; checks occur at different stages.

DefinitionInformationPrograms and processes

An operating system manages a program's execution as a process.

DefinitionInformationVirtual address spaces

A process's address view is distinct from physical memory placement.

DefinitionInformationFiles, directories and persistence

File contents, directory names and durable storage have distinct roles.

DefinitionMathematics & logicPolygon perimeter and parallel-base areas

Boundary length and enclosed area measure different properties of a plane shape.

MethodPhysics · ChemistryOpen, closed, isolated and adiabatic systems

A system boundary determines which transfers count as exchanges.

DefinitionPhysicsHypothesis, law and scientific theory

These terms name different scientific roles rather than steps in a promotion ladder.

DefinitionPhysicsConduction, convection and thermal radiation

Heat can be transferred by microscopic interactions, fluid motion or electromagnetic radiation.

DefinitionPhysicsConvective heat-transfer coefficient

The coefficient h represents a specified surface–fluid transfer situation.

ModelPhysicsLumped thermal relaxation and Biot criterion

An approximately uniform-temperature body can have an exponential thermal response.

ModelPhysicsNet radiation exchange and gray surfaces

Emission and absorption must both enter a net radiative heat balance.

ModelMeasurement · PhysicsSolid angle and steradian

A solid angle measures a cone of directions rather than a plane rotation.

Definition
Mathematical statementSource checked

Sine and cosine rules for triangles

Mathematics & logic

Side-angle laws beyond right triangles.

Key knowledge

In a nondegenerate Euclidean triangle, a/sin A=b/sin B=c/sin C, where each side faces its named angle. The cosine rule gives c²=a²+b²−2ab cos C. The corresponding formulas hold after consistent relabeling.

Values and equations

a/sin A=b/sin B=c/sin C; c²=a²+b²−2ab cos C

a, b, c
Side lengths opposite A, B, C respectively same length unit
A, B, C
Interior angles; C lies between sides a and b rad

Conditions and scope

  • Sides are positive lengths in a common unit; A,B,C are interior angles between 0 and π radians.
  • Two sides and a nonincluded angle can permit two triangles; inverse sine alone may miss the supplementary angle.
Sources · 2

References

Sources retain their recorded scope. A source link alone does not establish verification of every statement.

  1. TIMES: Further TrigonometryAustralian Mathematical Sciences InstituteCourse notesamsi.org.auRead source
  2. Mathematics XI · Sine and cosine rulesSiyavula, Everything MathsTeaching resourcesiyavula.comRead source

Evidence and review

Side-angle laws beyond right triangles.

2026-10-10 · Independent Physics and Measurement reference editor (AI editorial bounded review). Exact bounded introductory reference claims and this practical concept inventory are approved after closure. Existing university/research proof and source scopes remain separate; no universal knowledge-completeness or experimental-validation assertion.

Foundational review

2026-10-10 · Independent AI editorial inspection of this new compact reference entry, its equations, applicability, sources and knowledge connections.

Sources from other places

Agreement within this scope

Euclidean triangle sine and cosine rules, including a two-solution example.

Degrees versus radians do not change the relations when used consistently; the ambiguous example is not a complete classification.

Related reference entries

Explore relationships within this area of study and its connections to other areas.

Showing 1–2 of 2 relationships

Scientific relationships · 2 / 2

The relationship type describes the connection. Its linked statements, sources and conditions define the scope.

  1. Related to
    Relationship sources · 1

    Linked statements

    • In a nondegenerate Euclidean triangle, a/sin A=b/sin B=c/sin C, where each side faces its named angle. The cosine rule gives c²=a²+b²−2ab cos C. The corresponding formulas hold after consistent relabeling.

    At a right included angle the cosine correction vanishes.

  2. Uses definition
    Relationship sources · 1

    Linked statements

    • In a nondegenerate Euclidean triangle, a/sin A=b/sin B=c/sin C, where each side faces its named angle. The cosine rule gives c²=a²+b²−2ab cos C. The corresponding formulas hold after consistent relabeling.

    General triangle angles use the sine and cosine functions.

Entry detailsPermanent link to this entry ↗
Record identity
foundation-math-triangle-sine-cosine-rules
Revision
foundations-audit.1
Structured reference record
Foundational core completion