GRAPH OF THE SECANT FUNCTION

This section discusses the graph of the secant function (shown below).
For ease of reference, some material is repeated from the Trigonometric Functions.

one period of the graph of

$\displaystyle y = \sec x := \frac{1}{\cos x}$

(periodic with period $\,2\pi\,$)

The cosine curve is shown in red.

Key ideas contributing to the graph:
  • the reciprocal of $\,1\,$ is itself
  • the reciprocal of $\,-1\,$ is itself
  • the number $\,0\,$ has no reciprocal
  • the reciprocal of a small positive number
    is a large positive number
  • the reciprocal of a small negative number
    is a large negative number
  • a number smaller than $\,1\,$
    has reciprocal bigger than $\,1\,$

The Secant Function: Definition and Comments

Where does the graph of the secant function come from?

The graph of the secant function is easy to obtain as the reciprocal of the cosine function.
The key ideas are illuminated below:


the reciprocal of $\,1\,$ is $\,1\,$,
so the points shown
do not move

the reciprocal of $\,-1\,$ is $\,-1\,$,
so the points shown
do not move

zero has no reciprocal:
where the cosine is zero,
the secant has a vertical asymptote

the reciprocal
of a small positive number
is a large positive number

the reciprocal
of a small negative number
is a large negative number

the cosine curve is bounded
between $\,\color{red}{1}\,$ and $\,\color{red}{-1}\,$;
thus, the reciprocals all have
size greater than or equal to $\,\color{green}{1}\,$

Important Characteristics of the Graph of the Secant Function

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Review of Circles (and related concepts)
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