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.

The secant function \( \sec(\theta) \) is defined as:

\( \sec(\theta) = \dfrac{r}{x} \), where \( r = \sqrt{x^2+y^2} \).

This definition gives a direct relationship between \( \sec(\theta) \) and \( \cos(\theta) \):

\( \sec(\theta) = \dfrac{r}{x} = \dfrac{1}{\cos(\theta)} \).

Key Observations

• Periodicity: \( \sec(\theta+2\pi) = \sec(\theta) \), so the period is \( 2\pi \).
• Even Function: \( \sec(-\theta) = \sec(\theta) \), making the graph symmetric about the y-axis.

Using the unit circle, we can find \( \cos(\theta) \) and hence \( \sec(\theta) \) over one period \( [0, 2\pi] \):

Values at Quadrantal Angles

Behavior Near Asymptotes

As \( \theta \) approaches \( \frac{\pi}{2} \) from the left:

As \( \theta \) approaches \( \frac{\pi}{2} \) from the right:

Using limits:

• \( \lim_{\theta \to (\pi/2)^-} \sec(\theta) = \infty \)
• \( \lim_{\theta \to (\pi/2)^+} \sec(\theta) = -\infty \)

This behavior occurs at all \( x = \frac{\pi}{2} + n\pi \), where \( n \) is any integer.

Graph of \( y = \sec(x) \)

Note: The vertical asymptotes (dashed lines) occur at the zeros of \( \cos(x) \).

Properties of sec x

1. Period: \( 2\pi \)
2. Vertical Asymptotes: \( x = \frac{\pi}{2} + n\pi \), \( n \in \mathbb{Z} \)
3. Domain: All real numbers except \( x = \frac{\pi}{2} + n\pi \), \( n \in \mathbb{Z} \)
4. Range: \( (-\infty, -1] \cup [1, \infty) \)
5. Symmetry: Even function (symmetric about the y-axis)

Interactive Tutorial: General Form \( f(x) = a \sec(bx + c) + d \)

Explore how parameters affect the graph:

• Period: \( \dfrac{2\pi}{|b|} \)
• Phase Shift: \( -\dfrac{c}{b} \)

Exploration Questions

1. Set a=1, b=1, c=0, d=0. Note the period and asymptotes. How does changing 'a' affect the range?
2. Set a=1, c=0, d=0 and vary b. Compare the graph's period with \( \frac{2\pi}{|b|} \).
3. Set a=1, b=1, d=0 and increase c from 0. Observe the phase shift direction and compare with \( -\frac{c}{b} \).
4. Repeat with negative c values.
5. Test with b=2, 3, 4.
6. Set non-zero a,b,c and vary d. What shift occurs?
7. Which parameters affect asymptote positions? Explain analytically.
8. Which parameters affect the domain? Explain analytically.
9. Which parameters affect the range? Explain analytically.

Related Resources

• Cosine Function
• Graphs of Basic Trigonometric Functions
• Trigonometric Functions