Secant Calculator

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).