Secant Calculator
Calculate the secant of an angle in degrees, radians or gradians. The secant is the reciprocal of the cosine.
What is Secant?
Defined as the reciprocal of cosine, secant follows the relation sec(θ) = 1 / cos(θ). In a right triangle, it corresponds to the ratio of the hypotenuse to the adjacent side.
Among the notable values are sec(0°) = 1, sec(60°) = 2 and sec(45°) ≈ 1.4142. Wherever cosine reaches zero (90°, 270° and odd multiples of 90°), secant becomes undefined.
Unit conversions: 1° equals π/180 rad ≈ 0.01745 rad, while 1 rad equals 180°/π ≈ 57.296°. And 400 gradians correspond to 360°.
The secant function
The secant is the reciprocal of the cosine: sec(θ) = 1/cos(θ). Period is 2π and the vertical asymptotes coincide with the zeros of cosine. It is an even function: sec(−θ) = sec(θ). Reference values: sec(0) = 1, sec(30°) = 2√3/3, sec(45°) = √2, sec(60°) = 2, and sec(90°) undefined. A key identity is 1 + tan²θ = sec²θ, and the derivative is (sec x)' = sec x · tan x. The name comes from Latin secare, "to cut" — the same root as a secant line cutting through a circle.
Applications: integrals, arc length and optics
The secant features in the classical integral ∫ sec x dx = ln|sec x + tan x| + C, in arc-length formulas (the element ds = sec(θ) dx for a curve given as y(x) with slope tan(θ)), in Maclaurin series for trigonometric expansions, and in geometric optics where ray paths through inclined surfaces stretch by a factor of sec(θ) relative to the normal direction.
FAQ
Why is the period 2π and not π? Because cosine itself has period 2π, and taking the reciprocal does not change the period.
When is sec(θ) negative? Wherever cosine is negative — in the second and third quadrants, i.e. 90° < θ < 270° (excluding the asymptotes).
What is the minimum value? sec(θ) is unbounded above and below, but on the interval where it is positive, the minimum is sec(0) = 1.
Why the name "secant"? From the Latin verb secare, to cut: the geometric interpretation comes from a line that cuts through the unit circle, with length equal to 1/cos(θ).
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Calculate the secant of an angle
The secant is the inverse of the cosine, that is, 1 divided by the angle's cosine. It tends to appear in calculus, in integrals and in some physics and engineering applications. This calculator finds the secant of any angle in a direct way.
You enter the angle in degrees, radians or gradians, and the tool calculates the secant while handling the points where it doesn't exist (when the cosine is zero). Since it's one of the trigonometric functions the ordinary calculator rarely offers as a direct button, a dedicated tool saves the extra step of inverting the cosine by hand.
The calculation is done in the browser. It works for trigonometry and calculus studies, and for any application using the secant.