y = \sec x

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

The secant function $y = \sec x$ is the trigonometric function defined as the reciprocal of the cosine¹. In a right triangle it is the ratio of the hypotenuse to the adjacent side.

Definition

\sec x = \frac{1}{\cos x}

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

Domain and range

The domain is every real number with $x \neq \dfrac{\pi}{2} + n\pi$
The range is $y \leq -1$ or $y \geq 1$
The period is $2\pi$
It is an even function

Since $-1 \leq \cos x \leq 1$ and $\cos x \neq 0$ , we have $|\sec x| \geq 1$ , so no value strictly between $-1$ and $1$ is ever taken. The secant moves only outside the band that confined the cosine.

Symmetry and period

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

Asymptotes and limits

There is a vertical asymptote at each $x = \dfrac{\pi}{2} + n\pi$ .

Approach	$\cos x$	$\sec x$
$x \to \left( \dfrac{\pi}{2} \right)^{-}$	$0^{+}$	$+\infty$
$x \to \left( \dfrac{\pi}{2} \right)^{+}$	$0^{-}$	$-\infty$

Monotonicity and extrema

The derivative is $\dfrac{d}{dx}\sec x = \sec x \tan x$ . Between consecutive asymptotes the curve forms a U or an inverted U.

$x$	$\cos x$	$\sec x$	Extremum
$2n\pi$	$1$	$1$	local minimum
$(2n+1)\pi$	$-1$	$-1$	local maximum

Where the cosine is largest the secant is smallest, and where the cosine is smallest the secant is largest. Taking reciprocals exchanges large and small, so the closer the cosine comes to $0$ , the wilder the secant swings.

Relationships with other functions

It appears in one of the basic trigonometric identities.

1 + \tan^2 x = \sec^2 x

It is tied to the cosecant by $\sec x = \csc\left( x + \dfrac{\pi}{2} \right)$ , so the relation between sine and cosine carries over unchanged to the reciprocal side.

Integral

\int \sec x\,dx = \ln|\sec x + \tan x| + C

Applications

The stretching in the east-west direction of the Mercator projection
Calculations involving inclines and the refraction of light
Integrals in which $\sec^2 x$ appears as the derivative of the tangent

In the Mercator projection the east-west scale at latitude $\varphi$ is multiplied by $\sec\varphi$ ². Maps stretch so dramatically at high latitudes because this function grows so quickly.

Trigonometric functions, Wikipedia
Mercator projection, Wikipedia

Graph of the Secant Function y=sec⁡xy = \sec xy=secx