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trigonometric-functions

Abstract math for trigonometric functions: definitions, domains, ranges, key identities, and inverse functions. USE FOR: implementing sin/cos/tan/cot/sec/csc; implementing arcsin/arccos/arctan/atan2/acot/asec/acsc; validating input domains for inverse trig; applying Pythagorean and angle-sum identities; choosing atan vs atan2; precision near domain boundaries. DO NOT USE FOR: angle unit conversions (use angles-and-circular-arithmetic); coordinate system conversions (use coordinate-system-conversions); rotation representations (use 3d-rotation-theory).

日本語の概要は準備中です。原文の説明を表示しています。

NetFabric/NetFabric.Numericsスター 36(GitHub 36 + cccskills 0)

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  • references/formulas.md2.5 KB
  • references/numerical-stability.md2.3 KB

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Trigonometric Functions

Primary Functions

FunctionDefinitionDomainRange
sin θopposite / hypotenuseℝ[−1, 1]
cos θadjacent / hypotenuseℝ[−1, 1]
tan θsin θ / cos θℝ \ {π/2 + kπ}ℝ
cot θcos θ / sin θℝ \ {kπ}ℝ
sec θ1 / cos θℝ \ {π/2 + kπ}(−∞,−1] ∪ [1,∞)
csc θ1 / sin θℝ \ {kπ}(−∞,−1] ∪ [1,∞)

Inverse Functions

FunctionPrincipal RangeDomainNotes
arcsin x[−π/2, π/2][−1, 1]Odd function
arccos x[0, π][−1, 1]arccos = π/2 − arcsin
arctan x(−π/2, π/2)ℝOdd; atan(0) = 0
atan2(y,x)(−π, π]ℝ² \ {0,0}Signs of both args determine quadrant
acot x(0, π)ℝacot(x) = arctan(1/x) for x > 0
asec x[0,π] \ {π/2}|x| ≥ 1asec(x) = arccos(1/x)
acsc x[−π/2,π/2] \ {0}|x| ≥ 1acsc(x) = arcsin(1/x)

Key Identities

IdentityFormula
Pythagoreansin²θ + cos²θ = 1
Pythagorean (tan)1 + tan²θ = sec²θ
Pythagorean (cot)1 + cot²θ = csc²θ
Angle sum (sin)sin(α+β) = sin α cos β + cos α sin β
Angle sum (cos)cos(α+β) = cos α cos β − sin α sin β
Double angle (sin)sin(2θ) = 2 sin θ cos θ
Double angle (cos)cos(2θ) = cos²θ − sin²θ = 1 − 2sin²θ
Reflectionsin(π−θ) = sin θ; cos(π−θ) = −cos θ
Paritysin(−θ) = −sin θ (odd); cos(−θ) = cos θ (even)

atan2 vs atan

ScenarioUse
Need full 4-quadrant angle from (y, x)atan2(y, x)
x guaranteed positiveatan(y/x)
Computing angle between two vectorsatan2(cross, dot)

Reference Files

FileLoad When
references/formulas.mdFull identity table, reciprocal derivations, composition rules
references/numerical-stability.mdPrecision near ±1, atan2 singularities, cancellation in identities

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