Inverse Cosecant Calculator

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

⚠️ arccsc(x) is defined only for |x| ≥ 1.
x = 1 x = 2 x = √2 x = -1 x = -2 x = 10
arccsc(x) Results
arccsc(x) in degrees 30.00°
arccsc(x) in radians 0.523599
sin(θ) = 1/x 0.500000
cos(θ) 0.866025

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

Relation: arccsc(x) = arcsin(1/x).

Understanding the Inverse Cosecant (arccsc)

The inverse cosecant function, denoted arccsc(x) or csc⁻¹(x), returns the angle whose cosecant is x. Since cosecant 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): [-π/2, 0) ∪ (0, π/2]. That is, angles from –90° to 90° (excluding 0° where cosecant is undefined).

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

Relationship with Arcsine

A convenient way to compute arccsc(x) is:

arccsc(x) = arcsin(1 / x)

This works because if csc θ = x, then sin θ = 1/x, so θ = arcsin(1/x) (taking the principal value of arcsin, which lies in [-π/2, π/2], matches the principal range of arccsc).

Derivative and Integral

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

Common Values

x arccsc(x) (degrees) arccsc(x) (radians)
1 90° π/2
2/√3 ≈ 1.1547 60° π/3
√2 ≈ 1.4142 45° π/4
2 30° π/6
∞ 0° 0 (as a limit)
-2 -30° -π/6
-√2 -45° -π/4
-2/√3 -60° -π/3
-1 -90° -π/2

Tip: On the unit circle, the angle θ = arccsc(x) is shown in red. Its sine equals 1/x, and its cosine is √(1 – 1/x²) (always non‑negative because θ lies in [-π/2, π/2]).