Secant Function sec x

Definition and Graph of the secant Function

Let angle \( \theta \) be in standard position with initial side on the positive x axis and terminal side OM as shown below.

angle in standard position.
The secant function \( sec (\theta) \) is defined as
\( \sec(\theta) = \dfrac{r}{x} \) , where \( r \) is the distance from O to M and is given by \( r = \sqrt{x^2+y^2} \).
An examination of the definiton of the secant gives a relationship between \( sec (\theta) \) and \( cos (\theta) \) as follows
\( \sec(\theta) = \dfrac{r}{x} = \dfrac{1}{\cos(\theta)}\)

Note that
1) \( \sec(\theta+2\pi) = \dfrac{1}{\cos(\theta+2\pi)} = \dfrac{1}{\cos(\theta)}= \sec(\theta)\)
and therefore \( \sec(\theta) \) is a periodic function whose period is equal to \( 2\pi \).

2) \( \sec(-\theta) = \dfrac{1}{\cos(-\theta)} = \dfrac{1}{\cos(\theta)} = \dfrac{1}{\cos(\theta)} = \sec(\theta)\)
and therefore \( \sec(\theta) \) is an even function and its graph is symmetric with respect to the y axis.

We now use a unit circle to find \( \cos(\theta)\) and hence \( \sec(\theta)\) over one period extending from \( \theta = 0 \) to \( \theta = 2\pi \).
We know from the sine and cosine functions that the x and y coordinates on a unit circle gives the values of \( \sin(\theta)\) and \( \cos(\theta)\) as shown below.

unit circle to help read cos x.
Let us now put the values of the quadrantal angles angles \( 0, \dfrac{\pi}{2} , \pi , \dfrac{3\pi}{2} , 2\pi \) and the corresponding values of \( \cos(\theta)\) and \( \sec (\theta) = \dfrac{1}{\cos (\theta)} \) on a table as shown below.

\( \theta \) \( \cos(\theta) \) \( \sec (\theta) = \dfrac{1}{\cos (\theta)} \)
\( 0 \) \( 1 \) 1
\( \dfrac{\pi}{2} \) \( 0 \) undefined
\( \pi \) \( -1 \) -1
\( \dfrac{3\pi}{2} \) \( 0 \) undefined
\( 2\pi \) \( 1 \) 1

\( \sec(\theta)\) is undefined at \( \theta = \pi / 2 \) and \( \theta = 3\pi / 2 \) , however we can obtain information about the behaviour of \( \sec(\theta)\) close to these values using a claculator.
Use a calculator to find values of \( \sec(\theta)\) as \( \theta \) approaches \( \dfrac{\pi}{2} \approx 1.570796327 \) by values smaller than \( \dfrac{\pi}{2} \) starting at \( \theta = 1.500000 \)
\( \theta \) \( \sec(\theta) \)
\( 1.500000 \) \( 14.1368329 \)
\( 1.550000 \) \( 48.08888102 \)
\( 1.570000 \) \( 1255.76599 \)
\( 1.570700 \) \( 10381.32747 \)
\( 1.570791 \) \( 187730.1491\)
\( 1.570796 \) \( 3060023.307 \)

As \( \theta \) approaches \( \pi / 2 \) by values smaller than \( \pi / 2 \), \( \sec(\theta) \) approaches large values.

We now use the calculator to find values of \( \sec(\theta)\) as \( \theta \) approaches \( \pi / 2 \) by values larger than \( \pi / 2 \) starting at \( \theta = 1.580000 \)
\( \theta \) \( \sec(\theta) \)
\( 1.580000 \) \( -108.6538055 \)
\( 1.575000 \) \( -237.8878891 \)
\( 1.571000 \) \( -4909.826044 \)
\( 1.570800 \) \( -272241.8084 \)
As \( \theta \) approaches \( \pi / 2 \) by values larger than \( \pi / 2 \) , \( \sec(\theta) \) approaches small values and hence the existence of a vertical asymptote at \( \theta = \pi / 2 \).

Using the concept of limits, we describe the behaviour of \( \sec(\theta) \) as \( \theta \) approaches \( \pi / 2 \) from the left (or by values smaller than \( 0 \)) as follows
\( \lim_{\theta \to (\pi / 2)^-} \sec(\theta) = \infty \)
and the behaviour of \( \sec(\theta) \) as \( \theta \) approaches \( \pi / 2 \) from the right (or by values larger than \( \pi / 2 \)) as follows
\( \lim_{\theta \to (\pi / 2)^+} \sec(\theta) = -\infty \)
Similar behaviour occur close to all values of \( x = \pi / 2 + n\pi \) where \( n \) is an integer.

We now use a system of rectangular coordinates \( (x,y) \) to plot the points and use all information in the above tables to approximate the graph of the secant function \( \sec x \) as shown below.
\( \sec (\theta) \) is shown in red along with \( \cos (\theta) \) in green, from \( \theta = 0 \) to \( \theta = 2\pi \), to help understand the relationship, of the two functions over one period, graphically. The most important observation is: the vertical asymptotes (shown in broken lines) of \( \sec (\theta) \) occur at the position of the zeros of \( \cos (\theta) \).

NOTE
We are used to \( x \) being the variable of a function on the horizontal axis. Hence \(x\) on the graph takes values of \( \theta \) and y takes the values of \( \sec(\theta) \) which is noted as \( y = \sec(x) \). The broken vertical lines indicate the vertical asymptotes of \( \sec x \).


graph of sec(x) in a rectangular system of coordinates.

Properties of sec x

1) sec x has a period equal to \( 2\pi \).
2) \( \sec(x) \) has vertical asymptotes at all values of \( x = \pi / 2 + n\pi \) , \( n \) being any integer.
3) The domain of \( \sec(x) \) is the set of all real numbers except \( x = \pi / 2 + n\pi \) , \( n \) being any integer.
4) The range of \( \sec(x) \) is given by: \( (-\infty , -1] \cup [1, +\infty) \)
5) \( \sec(x) \) is an even function and its graph is symmetric with respect to y axis.

Interactive Tutorial on the secant Function sec x of the General Form

The secant function of the general form given by

\[ f(x) = a \sec(b x + c) + d \] and its properties such as period, phase shift, asymptotes domain and range are explored interactively using an app by changing the parameters a, b, c and d. The figure below shows an example of the graph of this function.
The period is given by: \( \dfrac{2\pi}{| b |} \)
The phase shift is given by: \( - c / b \)
graph of the secant function


Once you finish the present tutorial, you may want to go through a self test on
trigonometric graphs .

a =
b =
c =
d =
>

Click on the button "draw" above.
How change the values the 4 coefficients a,b,c and d affect the graph of f(x)?
  1. Set set a = 1, b = 1, c = 0 and d = 0. Write down f(x) and take note of the period, phase shift and positions of the asymptotes of f(x)? Now change a , how does it affect the range of the graph?
  2. Set a = 1, c = 0, d = 0 and change b. Find the period from the graph and compare it to \( \dfrac{2\pi}{| b |} \). How does b affect the graph of f(x)? How does it affect the asymptotes?
  3. Set a = 1, b = 1, d = 0 and change c starting from zero going slowly to positive large values. Take note of the shift, is it left or right, and compare it to \( - c/b \).
  4. Set a = 1, b = 1, d = 0 and change c starting from zero going slowly to negative smaller values. Take note of the shift, is it left or right, and compare it to \( - c/b \).
  5. repeat 3 and 4 above for b = 2 , 3 and 4.
  6. Set a, b and c to non zero values and change d. What is the direction of the shift of the graph?
  7. Which of the parameters affect the positions of the asymptotes? Explain analytically.
  8. Which of the parameters affect the domain of the secant function? Explain analytically.
  9. Which of the parameters affect the range of the secant function? Explain analytically.

More references and Links Related to the Secant function

Graph and properties of the cosine function. Cosine Function
Explore and analyze the graphs, the domains and range of sin(x), cos(x), tan(x), cot(x), sec(x) and sec(x).
Graphs of Basic Trigonometric Functions
Trigonometric Functions.