Graph of the Secant Function $y = \sec x$

The secant function $y = \sec x$ is the trigonometric function defined as the reciprocal of the cosine. In a right triangle it is the ratio of the hypotenuse to the adjacent side.

Definition

Because the denominator is the cosine, the value of the cosine appears directly as its reciprocal.

Domain and range

Where $\cos x = 0$, at $x = \dfrac{\pi}{2} + n\pi$ (with $n$ an integer), the denominator vanishes and the function is undefined. Its domain is therefore all real numbers except these points. Since $-1 \le \cos x \le 1$ and $\cos x \ne 0$, we have $|\sec x| \ge 1$, so the range is $y \le -1$ or $y \ge 1$. It never takes a value strictly between $-1$ and $1$.

Symmetry and period

Because $\cos x$ is even, $\sec(-x) = \sec x$, so $y = \sec x$ is an even function, symmetric about the $y$-axis. Its period is $2\pi$, the same as cosine.

Asymptotes and limits

There is a vertical asymptote at each $x = \dfrac{\pi}{2} + n\pi$. As $x$ approaches $\dfrac{\pi}{2}$ from the left, $\cos x \to 0^{+}$ so $\sec x \to +\infty$; from the right, $\cos x \to 0^{-}$ so $\sec x \to -\infty$.

Behavior and notable points

Between consecutive asymptotes the curve forms a $U$ shape or an inverted $U$. Where $\cos x$ attains its maximum $1$, at $x = 2n\pi$, $\sec x$ has a local minimum of $1$; where $\cos x$ attains its minimum $-1$, at $x = (2n+1)\pi$, $\sec x$ has a local maximum of $-1$. For example $\sec 0 = 1$ and $\sec \pi = -1$. Its derivative is $\dfrac{d}{dx}\sec x = \sec x \tan x$.

Relation to other functions

As the reciprocal of cosine, it relates to the cosecant by $\sec x = \csc\left(x + \dfrac{\pi}{2}\right)$ and appears in the identity $1 + \tan^2 x = \sec^2 x$.

Applications

The secant appears in integration, $\int \sec x\,dx = \ln|\sec x + \tan x| + C$, in the Mercator map projection, and in problems involving inclines and the refraction of light.