Graph reciprocal trigonometric functions and asymptotes

Ontario Grade 12 Mathematics

Trigonometric Functions

Use sine, cosine, and tangent to locate branches and vertical asymptotes

A reciprocal trigonometric graph is built from a familiar sine, cosine, or tangent graph. At each input, the reciprocal function's value is one divided by the related function's value. When that related value is zero, the reciprocal is undefined. These input values give vertical asymptotes. To sketch the graph, first use the related graph to locate zeros, signs, and useful points. Then draw the reciprocal branches around the asymptotes.

What you will learn

Recognize secant, cosecant, and cotangent as reciprocal trigonometric functions.
Use the zeros of sine, cosine, or tangent to locate reciprocal-function asymptotes.
Plot key points and sketch branches with the correct signs and repeating patterns.
Check a sketch by comparing it with its related trigonometric graph.

1. Prerequisite bridge: reciprocals and basic graphs

For a nonzero number, its reciprocal is one divided by that number. For example, the reciprocal of

2

is

\frac{1}{2}

. Zero has no reciprocal because division by zero is undefined.

You will use familiar features of the sine, cosine, and tangent graphs. Sine and cosine repeat every

2\pi

. Tangent repeats every

\pi

. A period is the horizontal length of one complete repeating cycle. Use radians for the angles in this lesson.

Sine and cosine have values from

-1

to

1

. Tangent has zeros at integer multiples of

\pi

and vertical asymptotes halfway between consecutive zeros. A zero is an input where a function's value is zero. These facts help locate the reciprocal graphs.

The expression

\frac{1}{0}

is undefined.

A reciprocal is defined only when the original value is nonzero.
Start with the related sine, cosine, or tangent graph.
Use the original graph's zeros, signs, and key values to build the reciprocal graph.

2. The three reciprocal trigonometric functions

The reciprocal of sine is cosecant, written

\csc x

. The reciprocal of cosine is secant, written

\sec x

. The reciprocal of tangent is cotangent, written

\cot x

. These are reciprocal functions, not inverse functions.

A vertical asymptote is a vertical line that a graph approaches as its values become very large positive or very large negative. For a reciprocal trigonometric function, asymptotes occur at inputs where the related function is zero. The reciprocal is undefined at those inputs.

At any input where the related function is

1

or

-1

, its reciprocal has the same value. If a related value is between

0

and

1

, its reciprocal is greater than

1

. If it is between

-1

and

0

, its reciprocal is less than

-1

. The reciprocal keeps the sign of the original value.

Since sine and cosine stay between

-1

and

1

, their reciprocals have magnitude at least

1

wherever they are defined. Thus secant and cosecant do not pass through the band between

y=-1

and

y=1

.

\csc x=\frac{1}{\sin x},\quad \sec x=\frac{1}{\cos x},\quad \cot x=\frac{1}{\tan x}

Cosecant is the reciprocal of sine.
Secant is the reciprocal of cosine.
Cotangent is the reciprocal of tangent.
The zeros of the related function give the reciprocal function's vertical asymptotes.

3. From zeros to asymptotes and branches

First mark the zeros of the related function. These input values become vertical asymptotes for its reciprocal. Next mark useful points where the related function is

1

or

-1

. The reciprocal has the same value at those points.

Sine is zero at

x=n\pi

, where

n

is any integer. Therefore, cosecant has vertical asymptotes at those inputs. Between consecutive asymptotes, sine keeps one sign. Cosecant keeps that sign too. At the sine maximum or minimum, the cosecant value is

1

or

-1

.

As a cosecant branch moves away from an asymptote toward its turning point, the magnitude of its values decreases toward

1

. Moving from the turning point toward an asymptote, the magnitude increases. The sign stays the same on that branch.

Cosine is zero at

x=\frac{\pi}{2}+n\pi

. Secant has vertical asymptotes at those inputs. Its branches pass through points where cosine is

1

or

-1

, so the corresponding secant values are also

1

or

-1

.

Tangent is zero at

x=n\pi

, so cotangent has vertical asymptotes there. Between consecutive asymptotes, cotangent decreases from large positive values to large negative values. Cotangent crosses zero where tangent has a vertical asymptote, at

x=\frac{\pi}{2}+n\pi

. When

\sin x=0

,

\csc x

is undefined.

Cosecant and cotangent have asymptotes at $x=n\pi$ .
Secant has asymptotes at $x=\frac{\pi}{2}+n\pi$ .
Secant and cosecant branches stay at or beyond $y=1$ or $y=-1$ .
Cotangent crosses the horizontal axis halfway between consecutive asymptotes.

4. Sketching and checking a graph

Use a steady order. Identify the related basic graph and its period. Mark the zeros of that graph as vertical asymptotes for the reciprocal. Then mark key points where its value is

1

or

-1

. Use the related graph's signs to decide whether each reciprocal branch is above or below the horizontal axis.

Draw asymptotes as dashed vertical guide lines. They are not part of the reciprocal graph because the function is undefined at those inputs. Draw a separate branch on each interval between asymptotes. Make each branch approach, but not touch, the guide lines.

A final check can catch common sketching errors. Verify that the asymptotes match the related function's zeros. Verify that every branch has the correct sign. Check that points where the related function is

1

or

-1

remain at those same values on the reciprocal graph.

Find zeros before drawing reciprocal branches.
Use signs from the related graph to place branches.
Keep branches separate at every vertical asymptote.
Check key points and repeating spacing against the related graph.

Key features of reciprocal trigonometric graphs

Function	Related function	Vertical asymptotes	Key feature
$\csc x$	$\sin x$	$x=n\pi$	Branches stay at or beyond $y=1$ or $y=-1$
$\sec x$	$\cos x$	$x=\frac{\pi}{2}+n\pi$	Branches stay at or beyond $y=1$ or $y=-1$
$\cot x$	$\tan x$	$x=n\pi$	Crosses zero at $x=\frac{\pi}{2}+n\pi$

Worked example

Sketching one period of cosecant

Sketch

y=\csc x

for

0<x<2\pi

. State its vertical asymptotes and mark the key points that shape its branches.

Find the asymptotes
Cosecant is the reciprocal of sine, so it is undefined wherever sine is zero. The sine zeros at the ends of this interval give the vertical asymptotes.
$x=0,\quad x=2\pi$
Locate key points
Sine reaches $1$ at $x=\frac{\pi}{2}$ and $-1$ at $x=\frac{3\pi}{2}$ . Taking these reciprocals leaves the values unchanged.
$\left(\frac{\pi}{2},1\right),\quad \left(\frac{3\pi}{2},-1\right)$
Draw the branches
Sine is positive between $0$ and $\pi$ , so cosecant is positive there. Its branch has a lowest point at $\left(\frac{\pi}{2},1\right)$ . Sine is negative between $\pi$ and $2\pi$ , so cosecant is negative there. Its branch has a highest point at $\left(\frac{3\pi}{2},-1\right)$ . On both branches, the magnitude decreases as the graph moves away from an asymptote toward the turning point.

Answer: The graph has vertical asymptotes at

x=0

and

x=2\pi

. It has a positive branch between

0

and

\pi

, with turning point

\left(\frac{\pi}{2},1\right)

, and a negative branch between

\pi

and

2\pi

, with turning point

\left(\frac{3\pi}{2},-1\right)

.

Check: The branch signs match sine on each interval. The reciprocal values at the sine maximum and minimum remain

1

and

-1

.

Common mistakes and how to avoid them

Putting an asymptote at a maximum or minimum of the related function.

Correction: An asymptote occurs where the related function is zero, because its reciprocal is undefined there.

Drawing secant or cosecant branches between

-1

and

1

.

Correction: Sine and cosine have magnitude at most

1

, so their reciprocals have magnitude at least

1

wherever defined.

Giving every branch the same sign.

Correction: A reciprocal has the same sign as the related function. Check the sign of sine, cosine, or tangent on each interval.

Connecting a branch through a vertical asymptote.

Correction: The function is undefined at an asymptote. Draw separate branches on its two sides.

Drawing a cosecant branch so that its magnitude grows as it moves away from an asymptote toward its turning point.

Correction: The magnitude decreases toward

1

as the branch moves away from an asymptote toward its turning point. It grows as the branch approaches an asymptote.

Lesson summary

Secant, cosecant, and cotangent are reciprocals of cosine, sine, and tangent.
Zeros of the related function become vertical asymptotes of its reciprocal.
Use reciprocal values and the related graph's signs to place key points and branches.
Check asymptote locations, signs, and key points before finalizing a sketch.

Check your understanding

Question 1

Where are the vertical asymptotes of

y=\sec x

?

$x=n\pi$
$x=\frac{\pi}{2}+n\pi$
$x=2n\pi$
$x=\frac{\pi}{2}+2n\pi$

Show answer and explanation

x=\frac{\pi}{2}+n\pi

Secant is the reciprocal of cosine. Cosine is zero at

x=\frac{\pi}{2}+n\pi

, so secant has vertical asymptotes at those inputs.

Question 2

What is the value of

\csc x

when

\sin x=-1

?

$1$
$-1$
$0$
It is undefined.

Show answer and explanation

-1

Cosecant is the reciprocal of sine. The reciprocal of

-1

is

-1

.

Question 3

On an interval where tangent is positive and nonzero, what can be said about cotangent?

It is positive.
It is negative.
It is zero throughout the interval.
It is undefined throughout the interval.

Show answer and explanation

It is positive.

Cotangent is the reciprocal of tangent. A nonzero positive value has a positive reciprocal.

Key terms

Reciprocal: For a nonzero number, its reciprocal is one divided by that number.
Vertical asymptote: A vertical line that a graph approaches as its values become very large positive or very large negative.
Period: The horizontal length of one complete repeating cycle of a graph.
Zero: An input where a function's value is zero.
Key point: A point that helps locate and shape a graph, such as a maximum or minimum.

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