How this instrument works
Secant carries its meaning in its own name: the word comes from the Latin secare, 'to cut.' Draw a unit circle with a tangent line touching it at the point where θ = 0°, then extend the ray for any angle θ until it crosses that tangent line — the length of that ray, measured from the circle's center, is exactly sec(θ). The line containing that ray cuts across the circle rather than grazing it, which is where the function borrows its name; the short piece of the tangent line the ray lands on, meanwhile, has length tan(θ), the reason that neighbouring function is named as it is.
In a right triangle, secant is hypotenuse over adjacent — the flip of cosine's adjacent over hypotenuse — and on the unit circle it is simply 1 divided by the x-coordinate cosine already supplies. That reciprocal relationship gets confused with an entirely different operation: arccos, cosine's inverse. Arccos takes a ratio between −1 and 1 and hands back an angle; secant takes an angle and hands back a number, cosine's own reciprocal, with no undoing involved. Reaching for a calculator's cos⁻¹ key expecting secant is the most common slip with this function, since the two share a symbol but compute nothing alike.
Because cosine itself never strays outside −1 to 1, its reciprocal can never land strictly between them: secant's output lives entirely in (−∞, −1] ∪ [1, ∞), reaching its smallest possible magnitude, exactly 1, at 0° and 180°, and racing toward infinity as θ closes in on 90° or 270°, where cosine itself touches zero. The same integral a calculus student meets, ∫sec(θ) dθ = ln|sec(θ) + tan(θ)|, turns out to be the formula behind the Mercator map projection, which spaces its lines of latitude using exactly that expression so a straight compass bearing on the chart matches a straight course on the globe.
- Enter your angle into the Angle, θ field — it reads in degrees by default, so 60 means 60°.
- Switch the field's unit selector to rad or turn if your figure is already in radians or full turns.
- Read sec(θ) for the result — cosine's reciprocal, computed at full precision.
- Set θ to 0° or 180° to watch sec(θ) settle at its smallest possible magnitude, exactly 1 or −1.
- Avoid θ values of 90° or 270° exactly — cosine is zero there and secant has nothing to return.
Worked example — secant of a 60° angle
Set Angle, θ to 60°, which the engine reads as 1.0471975511965976 radians. Cosine of that angle is 0.5 exactly, so sec(θ) = 1 ⁄ 0.5 returns 2.0 on the nose — sixty degrees is one of the handful of angles where the underlying cosine has a clean closed form, so the reciprocal comes out just as clean.
The neighbouring angles make good sanity checks. Drop θ to 0°, cosine's own peak of 1, and sec(θ) falls to its floor of exactly 1.0. Push θ to 120° instead, past the point where cosine turns negative, and sec(θ) returns −2.0 — the same magnitude as at 60°, since cos(120°) = −0.5, but carrying the sign cosine itself contributes to the reciprocal.
Questions
What is the formula for secant?
sec(θ) = 1 ⁄ cos(θ) — secant is defined as the reciprocal of cosine, nothing more. At θ = 60°, cos(θ) = 0.5, so sec(θ) = 1 ⁄ 0.5 = 2.0 exactly. Because it is a plain reciprocal, secant carries no unit of its own, just like cosine.
How is secant different from arccos, the inverse of cosine?
They solve opposite problems. Secant takes an angle and returns cosine's reciprocal, a number; arccos takes a number between −1 and 1 and returns the angle whose cosine produced it. Confusing the two is the most common mistake with this function — pressing a calculator's cos⁻¹ key expecting secant computes something else entirely.
Why is secant undefined at 90° and 270°?
Because cos(θ) equals exactly zero at each of those angles, and sec(θ) = 1 ⁄ cos(θ) has nothing to divide by there. Approach either angle from either side and secant's magnitude grows without bound rather than settling on a value — a genuine vertical asymptote, not a rounding artifact.
How does secant relate to tangent?
They satisfy the Pythagorean identity sec²(θ) = 1 + tan²(θ), which follows from dividing sin²(θ) + cos²(θ) = 1 through by cos²(θ). At θ = 60°, tan(60°) ≈ 1.7320508, and 1 + tan(60°)² ≈ 1 + 3.0 = 4, matching sec(60°)² = 2.0² = 4 exactly.
What is the range of secant, and why can't it equal 0.5?
Secant's output is restricted to (−∞, −1] ∪ [1, ∞), because cosine itself never leaves −1 to 1 and dividing 1 by a fraction smaller than 1 always yields something larger than 1 in magnitude. A value like 0.5 would require cos(θ) = 2, which no real angle produces.
Where does the secant function turn up outside a trigonometry class?
In the Mercator map projection, still the basis of most nautical and web maps: its vertical spacing between lines of latitude follows the integral of sec(θ), which is why a straight line drawn on a Mercator chart holds a constant compass bearing even though it curves on the actual globe. Secant also shows up in optics and structural work wherever a line-of-sight length is measured against an angle from the vertical.