Arcsecant Calculator

What is Arcsec?

Arcsec (arc secant) reverses the secant: starting from a value x, it gives you the angle θ for which sec(θ) = x. In practice you compute it as arcsec(x) = arccos(1/x).

The result falls within the interval [0°, 90°) ∪ (90°, 180°]. A few cases: arcsec(1) = 0°, arcsec(2) = 60°, arcsec(−1) = 180°.

Domain: x ≤ −1 or x ≥ 1. For values between −1 and 1 (exclusive), there is no real arcsec.

The arcsecant function

Arcsecant inverts the secant: asec(x) = θ whenever sec(θ) = x. Since |sec θ| ≥ 1 at every θ where the secant is defined, asec only accepts inputs from the union (−∞, −1] ∪ [1, ∞); feed it anything strictly between −1 and 1 and there is no real answer. The principal range works out to [0, π] \ {π/2}. Take asec(2) = π/3 = 60°, which checks out because sec(60°) = 1/cos(60°) = 2. One handy identity is asec(x) = acos(1/x), and it's your best bet for working the value out on a calculator with no asec key. The derivative comes to (asec x)' = 1/(|x|·√(x² − 1)).

Applications: calculus and theoretical contexts

Arcsecant leans more theoretical than practical, yet it does earn its keep in integration, where the antiderivative ∫ 1/(x·√(x² − 1)) dx = asec|x| + C is the form printed in many tables. You'll also meet it in optics when a refractive coefficient has to be turned into an angle, and in problem sets built to cover the whole family of inverse trigonometric functions.

FAQ

Why is the domain not all of ℝ? Start from sec θ = 1/cos θ. The cosine never exceeds 1 in absolute value, so the secant never drops below 1 in absolute value, and that leaves the values strictly between −1 and 1 forever out of reach.

How do I compute asec on a basic calculator? Lean on the identity asec(x) = acos(1/x). Flip x to its reciprocal, then run that through the inverse cosine.

Why is π/2 excluded from the range? At π/2 the cosine is zero, which makes sec(π/2) undefined, and an inverse can't map anything to a point the original function never produced.

Are there different range conventions? There are. A few authors carve up the range so the derivative formula can drop the absolute value, but [0, π] \ {π/2} lines up with asec(x) = acos(1/x) and remains the version you'll see most often.