Free Secant Calculator

Last updated September 23, 2026 · Report an error

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Secant is one of the three "reciprocal" trigonometric functions, defined as sec(θ) = 1/cos(θ). It shows up less often in introductory geometry than sine and cosine, but it's central to calculus (the derivative of tan(θ) is sec²(θ)), to surveying and optics calculations involving angles of incidence, and to the equations describing hyperbolic and elliptical orbits, where the secant of the true anomaly appears directly in the orbit equation. Because secant is undefined wherever cosine is zero — at 90°, 270°, and every 180° interval from there — this calculator flags those points explicitly instead of returning a nonsensical huge number.

How it works

Enter an angle and choose degrees or radians. The calculator first finds cos(θ) using the standard trigonometric definitions, then takes its reciprocal, 1/cos(θ), returning "undefined" instead of an error whenever cos(θ) is zero. Because secant is an even function (sec(-θ) = sec(θ)), the sign of your input angle never flips the sign of the result on its own — only the quadrant does.

Enter angle (θ).
Enter angle unit.
Click Calculate to see your results.

Examples

sec(60°)

cos(60°) = 0.5, so sec(60°) = 1/0.5 = 2.

sec(120°)

cos(120°) = -0.5, so sec(120°) = 1/(-0.5) = -2 — negative because 120° falls in the second quadrant, where cosine is negative.

sec(90°)

sec(90°) is undefined, because cos(90°) = 0 and division by zero has no value.

Common mistakes to avoid

Confusing secant with cosecant — secant is 1/cosine, cosecant is 1/sine.
Expecting a very large finite number at 90° instead of the correct "undefined" result.
Assuming secant's output can be any real number — it's restricted to (-∞,-1] ∪ [1,∞).
Mixing up sec⁻¹(x) (inverse secant, an angle) with 1/sec(x) (which just equals cos(x)).

Frequently asked questions

Secant is simply the reciprocal of cosine: sec(θ) = 1/cos(θ). Wherever cosine is close to 1 (near 0°), secant is close to 1 too; wherever cosine shrinks toward 0, secant grows without bound.

Secant never falls strictly between -1 and 1 — its output is always ≤ -1 or ≥ 1, since it's the reciprocal of a value between -1 and 1 (excluding 0).

No, and this is a very common mix-up. Arccosine (cos⁻¹) answers "which angle has this cosine value?" Secant answers "what is 1 divided by the cosine of this angle?" They are different operations entirely.

They're linked by the Pythagorean identity 1 + tan²(θ) = sec²(θ), which is why sec²(θ) shows up so often when integrating or differentiating tangent in calculus.

You can compute it that way, but a dedicated secant tool avoids re-deriving the reciprocal by hand and correctly flags undefined points instead of returning an extremely large (but technically finite) floating-point number.

Never — since sec(θ) = 1/cos(θ), it would require cos(θ) to be infinite, which never happens. Secant's graph has horizontal asymptote-free branches that only ever move away from zero.