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MathSecant Calculator - sec(x) and arcsec(x)
Compute sec(x) and arcsec(x) in degrees or radians. Reciprocal of cosine, unit-circle geometry, identity 1+tan²=sec², integral trick, real applications.
Inverse secant calculator
Have feedback? Report bugs, suggest features, or share your thoughts — we read them allWhat is the Secant Function?
The secant function, written sec(x), is one of the six trigonometric functions and the reciprocal of cosine. In a right triangle, sec(θ) is the ratio of the hypotenuse to the side adjacent to angle θ — exactly the inverse of cos(θ) = adjacent/hypotenuse. On the unit circle, sec(θ) is the length of the line segment from the origin to where the tangent at the angle's point meets the x-axis. The name itself comes from Latin secans (cutting), referring to a secant line on a circle.
Secant appears in calculus (especially the Pythagorean identity 1 + tan²(x) = sec²(x), used heavily in integration by trig substitution), in physics (the air mass through which sunlight passes is sec of the zenith angle, used in solar-energy and atmospheric optics), in optics (the lensmaker's equation in some derivations uses sec of incidence angle), and in computer graphics (the field-of-view scaling in 3D projection matrices uses 1/tan(fov/2) = cot(fov/2), with sec appearing in some related formulas).
Mathematical definition:
sec(x) = 1 / cos(x) = hypotenuse / adjacentKey properties of secant:
- Domain: sec(x) is defined for every real x except x = (2n+1)π/2 (90°, 270°, 450°, …), where cos(x) = 0 and the function blows up.
- Range: (−∞, −1] ∪ [1, +∞). Secant never takes a value strictly between −1 and 1, because cosine is bounded by ±1 and we are dividing 1 by it.
- Periodicity: sec(x) repeats every 2π radians (360°), the same as cosine.
- Even symmetry: sec(−x) = sec(x). The graph is mirror-symmetric across the y-axis, inheriting cosine's even symmetry. (This contrasts with cosecant, which is odd.)
- Vertical asymptotes: at x = π/2 + nπ where cosine equals zero. Between asymptotes the graph forms a U-shape (or upside-down U) with a single minimum or maximum at magnitude 1.
- Derivative: d/dx sec(x) = sec(x)·tan(x). Always defined wherever sec is defined.
Secant is the natural function whenever you have the hypotenuse and want a ratio over the adjacent side — the inverse problem of cosine, where you'd otherwise divide by cosine.
What is Inverse Secant (Arcsecant)?
The inverse secant, written arcsec(x) or sec⁻¹(x), takes a value with |x| ≥ 1 and returns the angle whose secant equals that value. It is the inverse operation of sec, restricted to a one-to-one canonical range so the inverse is well-defined.
Mathematical definition:
arcsec(x) = arccos(1/x), for |x| ≥ 1Key properties of the inverse secant:
- Domain: arcsec is defined only for |x| ≥ 1. For |x| < 1 no angle has that secant.
- Range: the canonical output is [0, π/2) ∪ (π/2, π] — angles between 0° and 180° excluding 90° (where sec is undefined). Some textbooks use a different range; check your source.
- Monotonicity: arcsec is strictly increasing on each branch of its domain. As x grows from 1 to ∞, arcsec(x) grows from 0° toward 90°; as x decreases from −1 to −∞, arcsec(x) grows from 180° toward 90°.
- Special values: arcsec(1) = 0 (0°), arcsec(2) = π/3 (60°), arcsec(√2) = π/4 (45°), arcsec(2/√3) = π/6 (30°), arcsec(−1) = π (180°).
- Derivative: d/dx arcsec(x) = 1 / (|x|·√(x² − 1)). The absolute value keeps the derivative positive on both branches.
Arcsecant is the function to use when you have a ratio of hypotenuse over adjacent and need to recover the angle — for instance, computing the zenith angle of the sun from a measured ratio of slant-path air mass to vertical air mass.
Common Secant Values
Important secant values for common angles:
- sec(0°) = 1 (minimum positive value)
- sec(30°) = 2/√3 ≈ 1.155
- sec(45°) = √2 ≈ 1.414
- sec(60°) = 2
- sec(90°) = undefined (vertical asymptote)
- sec(120°) = −2
- sec(135°) = −√2 ≈ −1.414
- sec(150°) = −2/√3 ≈ −1.155
Frequently Asked Questions
Because sec(x) = 1 / cos(x), and cos(90°) = 0. Division by zero is undefined, so sec(90°) — and sec(270°), sec(450°), sec((2n+1)π/2) for any integer n — has no value. Geometrically, on the unit circle sec(θ) is the x-intercept of the tangent line at the angle's point; when θ = 90° that point is (0, 1), the tangent is horizontal, and a horizontal line never crosses the x-axis. Approaching 90° from below, sec grows toward +∞: sec(89°) ≈ 57.30, sec(89.9°) ≈ 572.96. Approaching from above, it dives to −∞: sec(91°) ≈ −57.30. The graph of sec has a vertical asymptote at every odd multiple of π/2, exactly where cosine crosses zero. This is the same pattern as tangent's asymptotes — both functions die where cosine dies, since both have cos in the denominator.
Because cosine is bounded between −1 and +1, and secant is its reciprocal. If 0 < |cos(x)| ≤ 1, then |1/cos(x)| ≥ 1. So sec(x) always has magnitude at least 1. As cos(x) approaches 1 (its maximum), sec(x) approaches 1 from above; as cos(x) approaches 0 (its limit before becoming undefined), sec(x) shoots toward ±∞. The exact value sec(x) = 1 happens only at x = 2nπ (where cos = 1), and sec(x) = −1 only at x = π + 2nπ (where cos = −1). This bounded-by-1 gap is the visual signature of secant's graph: each branch is a U-shape (or upside-down U) whose tip touches exactly ±1 and whose arms shoot to infinity at the asymptotes. Cosecant has the same structure for the same reason. Tangent and cotangent, by contrast, sweep through every real value with no such gap.
Start from the master identity sin²(x) + cos²(x) = 1. Divide every term by cos²(x): tan²(x) + 1 = sec²(x), since sin²/cos² = tan² and 1/cos² = sec². This is one of the three Pythagorean identities — the other two are the original sin² + cos² = 1 and the cosecant version 1 + cot² = csc². The 1 + tan² = sec² identity is the powerhouse of integration by trigonometric substitution. When you see √(x² + 1) in an integrand, substituting x = tan(θ) turns it into √(tan²(θ) + 1) = √(sec²(θ)) = |sec(θ)|, eliminating the radical and reducing the integral to standard secant-tangent calculus. The same identity also makes the derivative of arctan come out to 1/(1 + x²) — a pleasingly clean formula that propagates throughout probability theory (the Cauchy distribution) and physics.
Derivative: d/dx sec(x) = sec(x)·tan(x). Proof: sec(x) = (cos(x))⁻¹, apply the chain rule: d/dx (cos(x))⁻¹ = −1·(cos(x))⁻² · (−sin(x)) = sin(x)/cos²(x) = (sin(x)/cos(x)) · (1/cos(x)) = tan(x)·sec(x). The two minus signs cancel, leaving a positive product of sec and tan. Integral: ∫sec(x) dx = ln|sec(x) + tan(x)| + C, equivalently ln|tan(x/2 + π/4)| + C. This is the classic 'magic multiplier' antiderivative: multiply numerator and denominator by (sec(x) + tan(x)), so the numerator becomes d/dx of the denominator, then substitute u = sec(x) + tan(x). The result, while not derivable by inspection, is the cornerstone of Mercator-projection cartography — the y-coordinate on a Mercator map is exactly ∫sec(θ) dθ from the equator to the latitude θ, which is why countries near the poles look so absurdly large.
(1) Atmospheric optics and solar engineering: the air mass that sunlight traverses to reach you is approximately sec(θ_zenith), where θ_zenith is the angle of the sun from straight overhead. At noon at the equator sec ≈ 1; at sunset sec → ∞, which is why sunsets are red — short-wavelength blue light is scattered out over the long path; (2) Mercator projection: the y-axis stretch factor at latitude φ is exactly sec(φ), which is why Greenland looks huge on world maps — the cumulative stretch from the equator to the North Pole is the integral ∫₀^(π/2) sec(φ) dφ, which diverges (the pole maps to infinity); (3) Calculus integration: trig substitution x = a·tan(θ) converts √(a² + x²) integrals into ∫sec(θ) dθ; (4) Surveying: slant distance = horizontal × sec(angle of elevation), useful when you measure horizontally and want the actual cable or beam length; (5) Particle physics: relativistic 'gamma factor' γ = 1/√(1 − v²/c²) can be written as sec(rapidity) in hyperbolic-trig formulations of special relativity.
Because secant's image is (−∞, −1] ∪ [1, +∞) — those are the only values it ever takes. Asking 'what angle has secant equal to 0.5?' is like asking 'what angle has cosine equal to 2?' — no such angle exists, since cos = 2 is outside cosine's range of [−1, 1]. Most calculators return NaN or an error for arcsec inputs in (−1, 1); some return complex values via the analytic continuation of arccos. For the principal real inverse, the rule is strict: |x| must be at least 1. The endpoints arcsec(1) = 0° and arcsec(−1) = 180° correspond to the unique inputs where sec attains its minimum (positive) and maximum (negative) reciprocal values. As |x| grows, the angle approaches 90° — but never reaches it, because sec has an asymptote there. So arcsec maps the disconnected domain [−∞, −1] ∪ [1, ∞] to the disconnected range [0, π/2) ∪ (π/2, π].
Critical question, frequently confused. sec(x) is the secant — the reciprocal of cosine, equal to 1/cos(x). cos⁻¹(x) is the inverse cosine function, also written arccos(x), which returns the angle whose cosine is x. The notation is the trap: when we write 'cos²(x)' we mean (cos(x))², so 'cos⁻¹(x)' looks like it should mean (cos(x))⁻¹ = 1/cos(x) = sec(x). But by convention cos⁻¹ means inverse function, not reciprocal. So cos⁻¹(0.5) = 60° (the angle), whereas sec(0.5) = 1/cos(0.5 rad) ≈ 1.139 (a ratio). To avoid this confusion, prefer arccos(x) and arcsec(x) for the inverse functions, and reserve sec(x) and cos⁻¹ for situations where context makes the meaning clear. Many programming languages exacerbate this — JavaScript's Math.acos is the arccos, but the −1 superscript on calculator buttons can mean either depending on the model.
Because cos(x) has period 2π and sec(x) = 1/cos(x). Reciprocating doesn't change the period — if a function repeats every 2π, so does its reciprocal (excluding the zeros, which become asymptotes). Verify: sec(x + 2π) = 1/cos(x + 2π) = 1/cos(x) = sec(x). The same holds for cosecant, which inherits sine's period 2π. By contrast, tangent and cotangent have the shorter period π because their definitions involve a sin/cos or cos/sin ratio that returns to the same value (with two sign flips that cancel) after a half-turn. So the four 'parent' functions (sin, cos, sec, csc) all have period 2π, and the two 'ratio' functions (tan, cot) have period π. This is why arcsec and arccsc have wider output ranges than arctan and arccot — they need to cover a full period rather than just a half.
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