TRIGONOMETRYTrigonometryMathematics Calculator
arcsec

The ArcSecant (Inverse Secant) Function

ArcSecant returns the angle whose secant equals the given value. arcsec(x) = arccos(1/x). Domain: |x| ≥ 1, Range: [0, π] excluding π/2. Essential for calculus integrals.

Concept Fundamentals
|x| ≥ 1
Domain
[0°, 180°] \ 90°
Range
arcsec(x) = arccos(1/x)
Definition
∫ dx/(x√(x²-1))
Integral

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arcsec(x) = arccos(1/x) — compute via arcsecant when the integral form suggests it. arcsec(-x) = π - arcsec(x) — reflection property; arcsec is neither odd nor even. d/dx[arcsec(x)] = 1/(|x|√(x²-1)) for |x| > 1.

Key quantities
|x| ≥ 1
Domain
Key relation
[0°, 180°] \ 90°
Range
Key relation
arcsec(x) = arccos(1/x)
Definition
Key relation
∫ dx/(x√(x²-1))
Integral
Key relation

Ready to run the numbers?

Why: ArcSecant appears in calculus: ∫ dx/(x√(x²-1)) = arcsec|x| + C. Used when integrating expressions involving secant.

How: arcsec(x) = arccos(1/x). Since sec(θ) = 1/cos(θ), the inverse gives the angle whose cosine is 1/x. Domain |x|≥1 ensures 1/x ∈ [-1,1].

arcsec(x) = arccos(1/x) — compute via arcsecant when the integral form suggests it.arcsec(-x) = π - arcsec(x) — reflection property; arcsec is neither odd nor even.
Sources:Wolfram MathWorld — Inverse SecantKhan Academy — Inverse Trig

Run the calculator when you are ready.

Start CalculatingEnter |x| ≥ 1 to find the angle whose secant equals it

Examples — Click to Load

arcsecant.sh
CALCULATED
$ arcsec --value 2 --output degrees
arcsec(x)
60°
arccsc(x)
30°
Ref Angle
60°
Domain
|x|≥1 ✓
sec(arcsec)
2
1/x
0.5
Range
[0°,180°]\\90°
Calc use
Integrals
Share:
ArcSecant Calculator Result
arcsec(2)
60°
Ref 60°sec(arcsec) = 21/x = 0.5
numbervibe.com/calculators/mathematics/trigonometry/arcsecant-calculator

Inverse Trig Value Breakdown

arcsec vs arccsc

arcsec vs arccsc

Calculation Breakdown

INPUT VALIDATION
Input Value
2
Domain Check
|x| >= 1 ✓
ext{arcsec} ext{defined} ext{only} ext{for} |x| geq 1
PRIMARY RESULT
Compute 1/x
0.5
ext{arcsec}(x) = ext{arccos}(1/x)
ARCSECANT RESULT
60°
arcsec(2) = arccos(0.5)
RELATED VALUES
Range
[0°, 180°] \ {90°}
ext{Excludes} \text{pi} /2 ext{where} ext{sec} ext{is} ext{undefined}
arccsc(x)
30°
ext{arccsc}(x) = ext{arcsin}(1/x)
Reference Angle
60°
VERIFICATION
sec(arcsec(x))
2
ext{Verification}: ext{equals} ext{input}

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

∫

∫ dx/(x√(x²-1)) = arcsec|x| + C — a standard calculus integral form.

— MIT OCW

↔️

arcsec(x) = arccos(1/x) — the inverse secant is defined via inverse cosine.

— Paul's Notes

Key Takeaways

  • • arcsec(x) = arccos(1/x). Domain: |x| ≥ 1, Range: [0, π] excluding π/2
  • • arcsec(1)=0°, arcsec(2)=60°, arcsec(√2)=45°, arcsec(-1)=180°. Less common than arcsin/arccos/arctan
  • • arcsec(-x) = π - arcsec(x). sec(arcsec(x)) = x for |x| ≥ 1
  • • Used in calculus integrals: ∫ dx/(x√(x²-1)) = arcsec|x| + C. Essential for trig substitution
  • • The range excludes π/2 because sec(θ) = 1/cos(θ) is undefined at θ = 90°

Did You Know?

📐arcsec appears in the integral ∫ dx/(x√(x²-1)) = arcsec|x| + C — a standard result in calculus textbooksSource: Paul's Notes
🔄arcsec(x) = arccos(1/x) is the computational definition. JavaScript has no native arcsec — use Math.acos(1/x)Source: MDN Web Docs
📊Less common than the main three inverse trig functions, but arcsec and arccsc complete the set of sixSource: Wolfram MathWorld
🔬d/dx[arcsec(x)] = 1/(|x|√(x²-1)) for |x| > 1. Used in differential equations and optimizationSource: MIT OCW
📡In hyperbolic geometry, arcsec relates to the Gudermannian function and Mercator projectionsSource: NIST
🎓Some calculus curricula use arcsec for ∫ dx/(x√(x²-a²)) substitution: x = a·sec(θ)Source: Khan Academy

How ArcSecant Works

Since sec(θ) = 1/cos(θ), we have arcsec(x) = arccos(1/x). The domain |x| ≥ 1 comes from the range of secant — sec never outputs values between -1 and 1.

Calculus Integrals

arcsec is less common in basic trig but essential in calculus. The integral ∫ dx/(x√(x²-1)) = arcsec|x| + C arises from trig substitution x = sec(θ).

Range Excludes π/2

The range [0, π] excludes π/2 because sec(π/2) = 1/cos(π/2) is undefined (division by zero). So arcsec never returns 90°.

Relation to arccsc

arcsec(x) and arccsc(x) are related: arccsc(x) = π/2 - arcsec(x) for x ≥ 1. Both use the reciprocal 1/x in their definitions.

Expert Tips

Always Check Domain

|x| ≥ 1 is required. For |x| < 1, arcsec is undefined. Use the ArcSecant Calculator to verify.

Compute via arccos

arcsec(x) = arccos(1/x). In code: Math.acos(1/x). Try the ArcCosine Calculator for the core function.

Trig Substitution

For ∫ dx/(x√(x²-a²)), use x = a·sec(θ). Then dx = a·sec(θ)tan(θ)dθ and √(x²-a²) = a·tan(θ).

Verify with sec(arcsec(x))

sec(arcsec(x)) = x for |x| ≥ 1. Use the Secant Calculator to confirm.

Inverse Trig Calculator Comparison

FeatureArcSecantArcCosecantArcCosine
Domain|x| ≥ 1|x| ≥ 1[-1, 1]
Range[0, π] \ {π/2}[-π/2, π/2] \ {0}[0, π]
Definitionarccos(1/x)arcsin(1/x)arccos(x)
Excludes90°0°None
Calculus useYesYesYes
Common useIntegralsIntegralsDot product
At x=10°90°0°
At x=260°30°N/A

Frequently Asked Questions

Why is arcsec only for |x| >= 1?

Secant = 1/cos never outputs values between -1 and 1. The range of sec is (-∞,-1] ∪ [1,∞), so arcsec has no meaning for |x| < 1.

How is arcsec related to arccos?

arcsec(x) = arccos(1/x). This identity is used for computation since most languages have acos but not asec. Always ensure |x| ≥ 1.

Why does the range exclude π/2?

sec(π/2) = 1/cos(π/2) is undefined (cos(90°)=0). So arcsec cannot return 90° — there is no x such that sec(θ)=x gives θ=90°.

Where is arcsec used in calculus?

∫ dx/(x√(x²-1)) = arcsec|x| + C. Also in trig substitution for integrals involving √(x²-a²) with x = a·sec(θ).

What is arcsec(-2)?

120° (or 2π/3 rad). arcsec(-2) = arccos(-0.5) = 120°. By reflection: arcsec(-x) = π - arcsec(x).

Does JavaScript have arcsec?

No. Use Math.acos(1/x) for arcsec(x). Ensure |x| ≥ 1 to avoid domain errors. This calculator handles that check.

How do arcsec and arccsc relate?

For x ≥ 1: arccsc(x) = π/2 - arcsec(x). They are complementary in a sense. arccsc(x) = arcsin(1/x).

When would I use arcsec in real applications?

Mainly in calculus (integration), differential equations, and some physics/engineering problems involving secant. Less common than arcsin/arccos/arctan in everyday use.

ArcSecant by the Numbers

|x| ≥ 1
Domain
[0°, 180°]
Range
Excl. 90°
Excludes
Calculus
Primary use

Disclaimer: This calculator provides results based on standard IEEE 754 floating-point arithmetic. Results are accurate to approximately 15 significant digits. Inputs with |x| < 1 are undefined for real arcsec. For calculus applications, verify integral forms with your textbook.

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