Topic 3.11

The Secant, Cosecant, and Cotangent Functions

In 3.9 we warned that $\sin ^{-1} \theta$ is not $1 \div \sin \theta$ . That reciprocal is useful enough to have its own name. This note introduces the three reciprocal trig functions and shows that each graph can be drawn directly from the graph you already know.

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Practice 33 problems Next: Chapter 3.12A Equivalent Representations of Trigonometric Functions

Three Reciprocal Functions

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

CONCEPT

On the unit circle

If the terminal ray of $\theta$ meets the unit circle at $(x, y)$ , then $\csc \theta = 1/y$ , $\sec \theta = 1/x$ , and $\cot \theta = x/y$ .

Each reciprocal is undefined exactly where its partner equals 0: $\csc \theta$ and $\cot \theta$ where $y = 0 (\theta = k\pi )$ , and $\sec \theta$ where $x = 0$ $(\theta = \pi /2 + k\pi )$ .

A reciprocal always has the same sign as its partner, so the quadrant sign rules from 3.2B still work.

COMMON MISTAKE

"Secant goes with sine, and cosecant goes with cosine."

It is the other way around: the "co" functions pair up. cosecant = 1 ÷ sine and secant = 1 ÷ cosine. One way to remember: every pair has exactly one "co" — sine & COsecant, COsine & secant, tangent & COtangent.

Worked example

Example 1. Find the exact values of $\sec (5\pi /6)$ , $\csc (7\pi /4)$ , $\cot (2\pi /3)$ , and $\sec (\pi /2)$ .

01
$\sec (5\pi /6)$ : $\cos (5\pi /6) = -\sqrt{3}/2$ (reference angle $\pi /6$ , Quadrant II). So $\sec (5\pi /6) = 1 \div (-\sqrt{3}/2) = -2/\sqrt{3} = -2\sqrt{3}/3$ .
02
$\csc (7\pi /4)$ : $\sin (7\pi /4) = -\sqrt{2}/2$ (Quadrant IV). So $\csc (7\pi /4) = -2/\sqrt{2} = -\sqrt{2}$ .
03
$\cot (2\pi /3)$ : the point is (−1/2, $\sqrt{3}/2$ ), so $\cot (2\pi /3) = x/y = (-1/2) \div (\sqrt{3}/2) = -1/\sqrt{3} = -\sqrt{3}/3$ .
04
$\sec (\pi /2)$ : $\cos (\pi /2) = 0$ , and $1 \div 0$ is undefined. So $\sec (\pi /2)$ does not exist.
A calculator has no $\csc$ , $\sec$ , or $\cot$ key, so use the reciprocals (in RADIAN mode): $\csc (0.4) = 1 \div \sin (0.4) \approx 2.568$ , $\sec (2.5) = 1 \div \cos (2.5) \approx -1.248$ , and $\cot (-1) = 1 \div \tan (-1) \approx -0.642$ .

COMMON MISTAKE

Finding $\csc (0.4)$ by pressing $\sin ^{-1}(0.4)$ and getting ≈ 0.412.

$\sin ^{-1}$ is the inverse function (3.9): it returns an angle. $\csc (0.4)$ is a reciprocal: compute $\sin (0.4)$ first, then take 1 ÷ that result.

COMMON MISTAKE

" $\tan (\pi /2)$ is undefined, so $\cot (\pi /2)$ is undefined too."

Use $\cot \theta = \cos \theta \div \sin \theta$ instead of $1 \div \tan \theta$ : $\cot (\pi /2) = 0 \div 1 = 0$ . Where tangent blows up, cotangent is 0, and where tangent is 0, cotangent blows up.

Quick check

If $\sin\theta = -\frac{3}{5}$ , what is $\csc\theta$ ?

Graphing a Reciprocal from Its Partner

The graph of $y = \csc \theta$ can be built point by point from $y = \sin \theta$ , using three facts about reciprocals:

sin θ	1	1/2	1/10	1/100	→ 0⁺
csc θ	1	2	10	100	→ ∞

CONCEPT

How a reciprocal graph behaves

Where the partner is 1 or −1, the reciprocal is also 1 or −1: the two graphs touch.

Where the partner gets close to 0, the reciprocal gets huge: a zero of the partner becomes a vertical asymptote.

Where the partner gets bigger, the reciprocal gets smaller, and the signs always match.

$y = \csc \theta$ : U-shaped branches that touch $y = \sin \theta$ at its maximum and minimum, with asymptotes at the zeros of sine.

$y = \sec \theta$ is built from $y = \cos \theta$ the same way: asymptotes at $\pi /2 + k\pi$ .

The branches of $\csc \theta$ open upward above $y = 1$ and downward below $y = -1$ . There are no outputs between −1 and 1, because sine never goes above 1 in size, so its reciprocal can never be smaller than 1 in size.

$y = \cot \theta$ : decreasing on each branch, asymptotes at $\theta = k\pi$ , zeros at $\theta = \pi /2 + k\pi$ .

Function	Period	Vertical asymptotes	Range
csc θ	2π	θ = kπ	y ≤ −1 or y ≥ 1
sec θ	2π	θ = π/2 + kπ	y ≤ −1 or y ≥ 1
cot θ	π	θ = kπ	all real numbers

Compare cotangent with tangent (3.8): both have period $\pi$ and range of all real numbers, but tangent increases while cotangent decreases, and their asymptotes and zeros trade places.

Quick check

Where does $y = \sec\theta$ have vertical asymptotes on $[0, 2\pi]$ ?

Transformations

To graph a transformed reciprocal function, graph its partner with the same constants first, as a dashed guide. Then apply the three facts from Section 2.

Worked example

Example 2. Graph $g(\theta ) = 2 \sec \theta + 1$ . State its vertical asymptotes and range.

01
Guide: $y = 2 \cos \theta + 1$ has midline $y = 1$ , amplitude 2, maximum 3 at $\theta = 0$ , and minimum −1 at $\theta = \pi$ (3.6A).
02
Asymptotes: $\sec \theta$ is undefined where $\cos \theta = 0$ , at $\theta = \pi /2 + k\pi$ . The constants $a = 2$ and $d = 1$ do not move them.
03
Branches touch the guide at its maximum and minimum: $(0, 3)$ opening upward and $(\pi , -1)$ opening downward.
04
Range: $y \le -1$ or $y \ge 3$ . Nothing lies between the guide's minimum and maximum.

The shaded band between −1 and 3 contains no outputs of $g$ .

COMMON MISTAKE

"The range of $2 \sec \theta + 1$ is $-1 \le y \le 3$ ."

That is the range of the GUIDE, $2 \cos \theta + 1$ . The reciprocal function lives outside that band: $\sec \theta$ is never between −1 and 1, so $2 \sec \theta + 1$ is never between −1 and 3.

Worked example

Example 3. Graph $h(\theta ) = -\csc 2\theta$ for $0 \le \theta \le \pi$ .

01
Guide: $y = -\sin 2\theta$ has period $\pi$ (3.6A) and is reflected: it starts at 0, goes DOWN to −1 at $\theta = \pi /4$ , and up to 1 at $\theta = 3\pi /4$ .
02
Asymptotes at the zeros of $\sin 2\theta$ : $2\theta = k\pi$ , so $\theta = k\pi /2$ . In $[0, \pi ]$ that means $\theta = 0$ , $\pi /2$ , and $\pi$ .
03
Touch the guide at $(\pi /4, -1)$ with a branch opening downward, and at $(3\pi /4, 1)$ with a branch opening upward. Because of the reflection, $(\pi /4, -1)$ is a local maximum of $h$ , not a minimum.

$h(\theta ) = -\csc 2\theta$ : two branches between consecutive asymptotes, each touching the guide.

The same method gives cotangent asymptotes: $y = \cot 3\theta$ is undefined where $\sin$ $3\theta = 0$ , so $3\theta = k\pi$ and $\theta = k\pi /3$ , with period $\pi /3$ .

REAL-LIFE EXAMPLE

A spotlight on a wall

A spotlight sits 10 m from a long wall and turns, making angle $\theta$ with the perpendicular line to the wall. In the right triangle, the 10 m side is adjacent to $\theta$ , so the beam's length to the wall is $L = 10 \div \cos \theta = 10 \sec \theta$ .

At $\theta = \pi /3$ the beam is $10 \sec (\pi /3) = 20$ m long, and it hits the wall $10 \tan (\pi /3) = 10\sqrt{3} \approx 17.3$ m from the nearest point.

As $\theta$ approaches $\pi /2$ the beam becomes parallel to the wall and never reaches it: at $\theta = 1.5$ it is already about 141.4 m long. That is the vertical asymptote of $\sec \theta$ at $\pi /2$ , in real life.

The beam length is a secant, and the distance along the wall is a tangent.

Common slips

"Secant goes with sine, and cosecant goes with cosine."

It is the other way around: the "co" functions pair up. cosecant = 1 ÷ sine and secant = 1 ÷ cosine. One way to remember: every pair has exactly one "co" — sine & COsecant, COsine & secant, tangent & COtangent.
Finding $\csc (0.4)$ by pressing $\sin ^{-1}(0.4)$ and getting ≈ 0.412.

$\sin ^{-1}$ is the inverse function (3.9): it returns an angle. $\csc (0.4)$ is a reciprocal: compute $\sin (0.4)$ first, then take 1 ÷ that result.
" $\tan (\pi /2)$ is undefined, so $\cot (\pi /2)$ is undefined too."

Use $\cot \theta = \cos \theta \div \sin \theta$ instead of $1 \div \tan \theta$ : $\cot (\pi /2) = 0 \div 1 = 0$ . Where tangent blows up, cotangent is 0, and where tangent is 0, cotangent blows up.
"The range of $2 \sec \theta + 1$ is $-1 \le y \le 3$ ."

That is the range of the guide, $2 \cos \theta + 1$ . The reciprocal function lives outside that band: $\sec \theta$ is never between −1 and 1, so $2 \sec \theta + 1$ is never between −1 and 3.

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9 cards · Secant, cosecant, cotangent

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Recap card

6 lines to re-read the night before.

01
$\csc \theta = 1/\sin \theta$ , $\sec \theta = 1/\cos \theta$ , $\cot \theta = 1/\tan \theta = \cos \theta /\sin \theta$ . Each pair has exactly one "co".
02
A reciprocal is undefined where its partner is 0, and has the same sign as its partner.
03
Graph a reciprocal from its partner: touch at $\pm 1$ , asymptotes at the partner's zeros, big where the partner is small.
04
$\csc$ and $\sec$ have period $2\pi$ and range $y \le -1$ or $y \ge 1$ ; $\cot$ has period $\pi$ , range all reals, and decreases on each branch.
05
For $a \sec \theta + d$ or $a \csc (b\theta ) + d$ , graph the sinusoidal guide first; the range is everything outside the guide's range.
06
Coming up in 3.12: identities that connect all six trig functions.

Practice 33 problems Next: Chapter 3.12A Equivalent Representations of Trigonometric Functions

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The Secant, Cosecant, and Cotangent Functions

Three Reciprocal Functions

Graphing a Reciprocal from Its Partner

Transformations

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