Secant, Cosecant, and Cotangent — AP® Precalculus

Part 1 · Hook

Why this matters

A guy wire runs from the top of a

12

meter pole to the ground at an angle

\theta

. Its length is

\frac{12}{\sin\theta}

. As the wire gets flatter, its length grows without bound. Reciprocals like

\frac{1}{\sin\theta}

come up so often that they have their own names.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. What is $\cos\frac{\pi}{3}$ ?

$\frac{1}{2}$
$\frac{\sqrt{3}}{2}$
$\frac{\sqrt{2}}{2}$
$0$

Show the answer

The angle $\frac{\pi}{3}$ meets the unit circle at $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$ , so its cosine is $\frac{1}{2}$ .

Correct: $\frac{1}{2}$ :
$\frac{\sqrt{3}}{2}$ :
$\frac{\sqrt{2}}{2}$ :
$0$ :

2. Where does $y = \tan x$ have vertical asymptotes?

Where $\cos x = 0$
Where $\sin x = 0$
Where $\tan x = 1$
Where $\cos x = 1$

Show the answer

$\tan x = \frac{\sin x}{\cos x}$ is undefined where its denominator, $\cos x$ , is $0$ .

Correct: Where $\cos x = 0$ :
Where $\sin x = 0$ :
Where $\tan x = 1$ :
Where $\cos x = 1$ :

Part 4 · See it

See it first

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. — Secant is the reciprocal of cosine: equal to it at $\pm 1$ , with asymptotes where cosine is $0$ . LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

Take reciprocals of $\cos$ , $\sin$ and $\tan$ $\sec x = \frac{1}{\cos x}$ , $\csc x = \frac{1}{\sin x}$ , $\cot x = \frac{\cos x}{\sin x}$
A denominator is $0$ a vertical asymptote: $\sec$ where $\cos x = 0$ , $\csc$ and $\cot$ where $\sin x = 0$
$\cos x$ or $\sin x$ equals $\pm 1$ the reciprocal equals it there: a turning point of $\sec$ or $\csc$
$|\cos x| \le 1$ and $|\sin x| \le 1$ $|\sec x| \ge 1$ and $|\csc x| \ge 1$ , so neither takes values strictly between $-1$ and $1$

Part 6 · Key ideas

Key ideas

Pair them by the letter they do not share: $\sec$ goes with $\cos$ , $\csc$ with $\sin$ .
$\sec$ has asymptotes where $\cos x = 0$ ; $\csc$ and $\cot$ where $\sin x = 0$ .
Where $\cos x$ decreases and stays positive, $\sec x$ increases: reciprocals move the opposite way.
$\sec$ and $\csc$ have period $2\pi$ and range $(-\infty, -1] \cup [1, \infty)$ ; $\cot$ has period $\pi$ and range $(-\infty, \infty)$ .

Part 7 · Misconception

A common mistake

The wrong idea: $\csc x$ is the reciprocal of $\cos x$ , because both start with c.

What actually happens: $\csc x = \frac{1}{\sin x}$ and $\sec x = \frac{1}{\cos x}$ . Each pairs with the function whose name it does not start like.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Graph

The graph of $y = \sec x$ , with $y = \cos x$ dashed

1. Using the graph or the unit circle, find the exact value of $\sec\frac{2\pi}{3}$ .

No calculator. Type the exact value (a fraction, π, e or a root is fine).

Show the answer

$\sec\frac{2\pi}{3}$ $= \frac{1}{\cos(2\pi/3)}$ $= \frac{1}{-1/2}$ $= -2$ . The graph of secant is below $-1$ on $\left(\frac{\pi}{2}, \frac{3\pi}{2}\right)$ , where cosine is negative.

Answer: -2

2. What is the range of $y = \sec x$ ?

$(-\infty, -1] \cup [1, \infty)$
$[-1, 1]$
$(-\infty, \infty)$
$[1, \infty)$

Show the answer

$\cos x$ is between $-1$ and $1$ , so its reciprocal has absolute value at least $1$ : $\sec x \ge 1$ where cosine is positive and $\sec x \le -1$ where it is negative. On the graph, no part of the secant curve lies strictly between $-1$ and $1$ .

Correct: $(-\infty, -1] \cup [1, \infty)$ : Right: the reciprocal of a number between $-1$ and $1$ is at least $1$ in size.
$[-1, 1]$ : That is the range of cosine; secant takes the reciprocals of those values.
$(-\infty, \infty)$ : Secant never takes values like $0$ or $\frac{1}{2}$ : $\frac{1}{\cos x} = \frac{1}{2}$ would need $\cos x = 2$ .
$[1, \infty)$ : Where $\cos x < 0$ , the secant branches open downward, with outputs at most $-1$ .

3. Using the graph, which statement about $\sec x$ near $x = \frac{\pi}{2}$ is true?

$\lim_{x \to \pi/2^-} \sec x$ $= \infty$
$\lim_{x \to \pi/2^-} \sec x$ $= -\infty$
$\lim_{x \to \pi/2^-} \sec x$ $= 0$
$\lim_{x \to \pi/2^-} \sec x$ $= 1$

Show the answer

Just left of $\frac{\pi}{2}$ , $\cos x$ is a small positive number, so $\frac{1}{\cos x}$ is a large positive number: $\lim_{x \to \pi/2^-} \sec x$ $= \infty$ . From the right, $\cos x$ is small and negative, and the limit is $-\infty$ .

Correct: $\lim_{x \to \pi/2^-} \sec x$ $= \infty$ : Right: the reciprocal of a small positive number is large and positive.
$\lim_{x \to \pi/2^-} \sec x$ $= -\infty$ : That is the behavior from the right, where cosine is negative.
$\lim_{x \to \pi/2^-} \sec x$ $= 0$ : Cosine approaches $0$ ; its reciprocal grows without bound instead.
$\lim_{x \to \pi/2^-} \sec x$ $= 1$ : $\sec 0 = 1$ is the bottom of the branch, not its behavior near the asymptote.

Table

Sine and cosine of four angles

Exact values of sin θ and cos θ for selected angles θ, in radians
$\theta$ (radians)	$\frac{\pi}{6}$	$\frac{\pi}{4}$	$\frac{\pi}{3}$	$\frac{\pi}{2}$
$\sin\theta$	$\frac{1}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$	$1$
$\cos\theta$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{1}{2}$	$0$

4. Using the table, find the exact value of $\csc\frac{\pi}{4}$ .

No calculator. Type the exact value (a fraction, π, e or a root is fine).

Show the answer

$\csc\frac{\pi}{4}$ $= \frac{1}{\sin(\pi/4)}$ $= \frac{1}{\sqrt{2}/2}$ $= \frac{2}{\sqrt{2}}$ $= \sqrt{2}$ .

Answer: $\sqrt{2}$ ≈ 1.414

5. Rewrite $\cos x \csc x$ as a single trigonometric function of $x$ .

Type an expression for $\cos x \csc x =$ .

Show the answer

$\cos x \csc x$ $= \cos x \cdot \frac{1}{\sin x}$ $= \frac{\cos x}{\sin x}$ $= \cot x$ .

Answer: $\cos x \csc x =$ $\cot(x)$

Part 9 · Summary

Summary

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

,

\csc x = \frac{1}{\sin x}

and

\cot x = \frac{\cos x}{\sin x}

. Each has vertical asymptotes where its denominator is

0

. Secant and cosecant equal their partners at

\pm 1

and never take values strictly between

-1

and

1

; cotangent takes every real value and has period

\pi

.

Part 10 · Up next

What comes next

Part 11 · Connections

Why this matters

What this builds on

Quick check before you start

See it first

How it works, step by step

Key ideas

A common mistake

Check yourself

The graph of y=sec⁡xy = \sec x, with y=cos⁡xy = \cos x dashed

Sine and cosine of four angles

Summary

What comes next

Connections

The graph of $y = \sec x$ , with $y = \cos x$ dashed