Introduction

The geometric construction of the secant from the unit circle is developed in secant and cosecant. Here, the secant is treated as a real function of a real variable.

The secant function $f(x) = \sec(x)$ assigns to each angle $x,$ measured in radians, the reciprocal of its cosine value, defined wherever $\cos(x) \neq 0.$ Its graph is a periodic curve with period $2\pi$ and has vertical asymptotes where the cosine of $x$ vanishes, at $x = \pi/2 + k\pi$ with $k \in \mathbb{Z}.$ The domain is the set of all real numbers except these points, and the range is $(-\infty, -1] \cup [1, +\infty).$

3π/2 -3π/2 The secant function is characterized by an infinite, periodically repeating family of vertical asymptotes throughout its domain. x 1 -1 -π/2 π/2 y

The secant is the reciprocal of the cosine, so it stays bounded where the cosine is near $\pm 1$ and grows without bound as the cosine approaches zero:

$$\sec(x) = \frac{1}{\cos(x)}$$

Because the cosine never exceeds $1$ in absolute value, its reciprocal never falls between $-1$ and $1,$ and each branch reaches a single extremum of $1$ or $-1$ where the cosine equals $\pm 1$ before diverging toward the neighbouring asymptotes.

Properties

The following properties of the secant function follow from its definition as the reciprocal of the cosine.

  • Domain: $\{\ x \in \mathbb{R} \mid x \neq \frac{\pi}{2} + k\pi \ \text{ for all } k \in \mathbb{Z} \ \}$
  • Range: $y \in (-\infty, -1] \cup [1, +\infty)$
  • Periodicity: periodic in $x$ with period $2\pi$
  • Parity: even, with $\sec(-x) = \sec(x)$
  • The graph has vertical asymptotes at $x = \frac{\pi}{2} + k\pi$ with $k \in \mathbb{Z}$

Relation with the tangent

The secant and the tangent are tied by a Pythagorean identity. Dividing $\sin^2(x) + \cos^2(x) = 1$ by $\cos^2(x)$ and using the reciprocal definitions gives:

$$\sec^2(x) = 1 + \tan^2(x)$$

The two functions share the same vertical asymptotes, and the secant grows together with the tangent.

Limits, derivatives, and integrals of the secant function

Near the origin the cosine reaches its maximum, so the secant takes its least positive value:

$$\lim_{x \to 0} \sec(x) = 1$$

The behaviour near the first vertical asymptote is described by one-sided limits. As $x$ approaches $\pi/2$ from the left the cosine is positive and tends to zero, so the function grows without bound,

$$\lim_{x \to \frac{\pi}{2}^-} \sec(x) = +\infty$$

while from the right the cosine is negative and the values diverge to negative infinity,

$$\lim_{x \to \frac{\pi}{2}^+} \sec(x) = -\infty$$

The function is continuous and differentiable on its domain. Its derivative is:

$$\frac{d}{dx}\sec(x) = \sec(x)\tan(x)$$

The indefinite integral is:

$$\int \sec(x) \ dx = \ln\left|\sec(x) + \tan(x)\right| + c$$

A broader treatment of trigonometric integrals, with the transformation and substitution techniques for the more complex cases, is given in trigonometric function integrals.

The secant function can also be written using imaginary numbers. With $e^{ix}$ the exponential function of base $e$ and $i$ the imaginary unit, Euler's formula gives:

$$\sec(x) = \frac{2}{e^{ix} + e^{-ix}}$$