Secant, Cosecant, and Cotangent — AP® Precalculus Notes

Three more trigonometric functions come from the ones you know by taking reciprocals. They are not new ideas, just names for quotients that keep appearing: the length of a wire, the distance along a sloped beam, the rewriting steps in the next topic. Knowing them by sight makes the algebra shorter.

The definitions

For any angle $x$ where the denominator is not $0$ :

$\sec x = \frac{1}{\cos x} \qquad \csc x = \frac{1}{\sin x} \qquad \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x}$

Secant goes with cosine and cosecant goes with sine. A quick check: each pairs with the function whose name it does not start like. Cotangent is the reciprocal of tangent, so it is cosine over sine.

Worked example 1. Find $\sec\frac{\pi}{4}$ , $\csc\frac{7\pi}{6}$ and $\cot\frac{2\pi}{3}$ .

$\cos\frac{\pi}{4} = \frac{\sqrt{2}}{2}$ , so $\sec\frac{\pi}{4} = \frac{2}{\sqrt{2}} = \sqrt{2}$ .

$\sin\frac{7\pi}{6} = -\frac{1}{2}$ , so $\csc\frac{7\pi}{6} = -2$ . The sign carries over: a reciprocal has the same sign as the number.

$\cos\frac{2\pi}{3} = -\frac{1}{2}$ and $\sin\frac{2\pi}{3} = \frac{\sqrt{3}}{2}$ , so $\cot\frac{2\pi}{3}$ $= \frac{-1/2}{\sqrt{3}/2}$ $= -\frac{1}{\sqrt{3}}$ $= -\frac{\sqrt{3}}{3}$ .

Graphs from reciprocals

You can sketch $y = \sec x$ straight from $y = \cos x$ (Figure 1):

Where $\cos x = 1$ or $-1$ , $\sec x$ equals it: the reciprocal of $\pm 1$ is $\pm 1$ . These are the turning points of the secant branches.
Where $\cos x = 0$ , $\sec x$ is undefined: a vertical asymptote. Near it, $\cos x$ is a tiny number, so $\sec x$ is huge, with the same sign as $\cos x$ . For example $\lim_{x \to \pi/2^-} \sec x$ $= \infty$ and $\lim_{x \to \pi/2^+} \sec x$ $= -\infty$ .
Where $\cos x$ is positive and decreasing, $\sec x$ is positive and increasing. Reciprocals of positive numbers move the opposite way.

The graph of y = sec x for x from 0 to 2 pi, with the dashed cosine curve. Secant equals cosine where cosine is 1 or -1, has dashed vertical asymptotes at pi/2 and 3 pi/2 where cosine is 0, and never takes values between -1 and 1. — **Figure 1.** $y = \sec x$ (solid) and $y = \cos x$ (dashed). LevlPrep original diagram.

The graph of $\csc x$ is built from $\sin x$ in the same way: turning points where $\sin x = \pm 1$ , asymptotes where $\sin x = 0$ . The graph of $\cot x$ looks like a tangent graph reflected and shifted: it falls on each branch, has zeros where $\cos x = 0$ and asymptotes where $\sin x = 0$ .

All six side by side

Function	Definition	Period	Vertical asymptotes	Range
$\sin x$	$y$ -coordinate	$2\pi$	none	$[-1, 1]$
$\cos x$	$x$ -coordinate	$2\pi$	none	$[-1, 1]$
$\tan x$	$\frac{\sin x}{\cos x}$	$\pi$	$x = \frac{\pi}{2} + k\pi$	$(-\infty, \infty)$
$\csc x$	$\frac{1}{\sin x}$	$2\pi$	$x = k\pi$	$(-\infty, -1] \cup [1, \infty)$
$\sec x$	$\frac{1}{\cos x}$	$2\pi$	$x = \frac{\pi}{2} + k\pi$	$(-\infty, -1] \cup [1, \infty)$
$\cot x$	$\frac{\cos x}{\sin x}$	$\pi$	$x = k\pi$	$(-\infty, \infty)$

Why the ranges skip $(-1, 1)$ : $|\cos x| \le 1$ , and dividing $1$ by a number of size at most $1$ gives a number of size at least $1$ .

Rewriting with sines and cosines

When an expression mixes the six functions, write everything in terms of $\sin x$ and $\cos x$ , simplify the fraction, then name what is left. This is the main skill the next topic builds on.

Worked example 2. Rewrite $\sin x \sec x$ and $\frac{\csc x}{\cot x}$ as single trigonometric functions.

$\sin x \sec x$ $= \sin x \cdot \frac{1}{\cos x}$ $= \frac{\sin x}{\cos x}$ $= \tan x$ .

$\frac{\csc x}{\cot x}$ $= \frac{1/\sin x}{\cos x/\sin x}$ $= \frac{1}{\sin x} \cdot \frac{\sin x}{\cos x}$ $= \frac{1}{\cos x}$ $= \sec x$ , wherever $\sin x \ne 0$ and $\cos x \ne 0$ .

Common slips

Pairing $\csc$ with $\cos$ . Cosecant is $\frac{1}{\sin x}$ .
Confusing the reciprocal with the inverse. $\sec x = \frac{1}{\cos x}$ returns a number; $\cos^{-1} x$ returns an angle.
Putting the asymptotes of $\sec x$ where $\sin x = 0$ . They come from the zeros of its own denominator, $\cos x$ .