How do you find the domain & range for sec theta?

Mar 5, 2016

Domain

$\left\{x | x = \left(k + \frac{1}{2}\right) \pi , k \in \mathbb{R} \setminus \setminus \setminus \mathbb{Z}\right\}$

Range

$\left(- \infty , 1\right] \cup \left[1 , \infty\right)$

Explanation:

Graph of $y = \sec \left(x\right)$
graph{sec(x) [-20, 20, -10, 10]}
$\sec \theta = \frac{1}{\cos} \theta$. As usual, division of zero is not allowed.

$\cos \theta = 0$ when $\theta = \frac{\pi}{2} , \frac{3 \pi}{2} , \frac{5 \pi}{2} \ldots$

In general, $\cos \theta = 0$ when $\theta = \left(k + \frac{1}{2}\right) \pi$, for $k \in \mathbb{Z}$.

The domain for $\sec \left(\theta\right)$ is any real number that

when subtracted $\frac{\pi}{2}$, is not an integer multiple of $\pi$.

In mathematical notations, it is

$\left\{x | x = \left(k + \frac{1}{2}\right) \pi , k \in \mathbb{R} \setminus \setminus \setminus \mathbb{Z}\right\}$

Note that the domain of $\sec \left(\theta\right)$ and $\tan \left(\theta\right)$ are identical.

The since $- 1 \le \cos \left(\theta\right) \le 1$, you can look at the graph of $y = \frac{1}{x}$, and close in on the portion of $- 1 \le x \le 1$.
graph{1/x [-5, 5, -2.5, 2.5]}
You will see that it is either $y \ge 1$ or $y \le - 1$. Similarly, for $\sec \left(\theta\right)$, it is either $\sec \left(\theta\right) \ge 1$ or $\sec \left(\theta\right) \le - 1$.

In mathematical notations, it is

$\left(- \infty , 1\right] \cup \left[1 , \infty\right)$