Inverse Secant Calculator

Compute the principal value of arcsec(x) for any real x with |x| ≥ 1. Visualize on the unit circle and explore the properties.

⚠️ arcsec(x) is defined only for |x| ≥ 1.
x = 1 x = 2 x = √2 x = -1 x = -2 x = 10
arcsec(x) Results
arcsec(x) in degrees 60.00°
arcsec(x) in radians 1.047198
cos(θ) = 1/x 0.500000
sin(θ) 0.866025

Definition: arcsec(x) = θ such that sec θ = x and θ ∈ [0, π/2) ∪ (π/2, π] (principal value).

Relation: arcsec(x) = arccos(1/x).

Understanding the Inverse Secant (arcsec)

The inverse secant function, denoted arcsec(x) or sec⁻¹(x), returns the angle whose secant is x. Since secant is not one‑to‑one over its whole domain, we restrict its range to obtain a principal value.

Domain and Range

  • Domain: |x| ≥ 1 (all real numbers with absolute value at least 1).
  • Range (principal value): [0, π/2) ∪ (π/2, π]. That is, angles from 0 to π (excluding π/2 where secant is undefined).

In this range, the secant function is strictly increasing on each piece, ensuring a unique inverse.

Relationship with Arccosine

A convenient way to compute arcsec(x) is:

arcsec(x) = arccos(1 / x)

This works because if sec θ = x, then cos θ = 1/x, so θ = arccos(1/x) (taking the principal value of arccos, which lies in [0,π], matches the principal range of arcsec).

Derivative and Integral

  • Derivative: d/dx [arcsec(x)] = 1 / (|x| √(x² – 1)) for |x| > 1.
  • Indefinite integral: ∫ arcsec(x) dx = x arcsec(x) – sign(x) · ln|x + √(x² – 1)| + C (for |x| > 1).

Common Values

x arcsec(x) (degrees) arcsec(x) (radians)
1 0° 0
2/√3 ≈ 1.1547 30° π/6
√2 ≈ 1.4142 45° π/4
2 60° π/3
∞ 90° π/2 (as a limit)
-2 120° 2π/3
-√2 135° 3π/4
-2/√3 150° 5π/6
-1 180° π

Tip: On the unit circle, the angle θ = arcsec(x) is shown in red. Its cosine equals 1/x, and its sine is √(1 – 1/x²) (positive for θ in [0,π] except at endpoints).