Q. 11 - Precalculus Enhanced with Graphing Utilities

Question

For what numbers $x, -2\mathrm{\pi }\le \mathrm{x}\le 2\mathrm{\pi },$ does $secx=1$? For what numbers x does $secx=-1?$

Step-by-Step Solution

Answer

For $x=-2\mathrm{\pi }, 0, 2\mathrm{\pi }$, $secx=1$ and for $x=-\mathrm{\pi },\mathrm{\pi },$ $secx=-1.$

Step 1. Given Information

We have to find for what numbers do $secx=1$ and $secx=-1.$

Step 2. Finding the numbers

To determine the numbers of the function where $secx=1$ and $secx=-1$at $-2\mathrm{\pi }\le \mathrm{x}\le 2\mathrm{\pi }$. We will substitute some values for x  to get corresponding values of y.

So, let $y=secx and x=-2\mathrm{\pi }$

$$\begin{gathered} y=secx \\ y=sec\left(-2\mathrm{\pi }\right) \\ y=1 \end{gathered}$$

Then let $x=-\mathrm{\pi }$

$$\begin{gathered} y=sec\left(-\mathrm{\pi }\right) \\ y=-1 \end{gathered}$$

Let $x=0$

$$\begin{gathered} y=sec\left(0\right) \\ y=1 \end{gathered}$$

Let $x=\mathrm{\pi }$

$$\begin{gathered} y=secx \\ y=sec\left(\mathrm{\pi }\right) \\ y=-1 \end{gathered}$$

Now, let $x=2\mathrm{\pi }$

$$\begin{gathered} y=sec\left(2\mathrm{\pi }\right) \\ y=1 \end{gathered}$$

Thus, for $x=-2\mathrm{\pi },0,2\mathrm{\pi }$, $secx=1$ and for $x=-\mathrm{\pi },\mathrm{\pi },$ $secx=-1.$