Chapter 6.7, Problem 9CFU

To determine

To graph: the trigonometric function.

Explanation of Solution

Given information:

  $y=\sec \left(2\theta +\pi \right)-1$

Calculation:

The given function is in the form of $y=A\sec \left(kx+c\right)+h$ . Compare the given function with the cosine function to get $x=\theta ,A=1,k=2,c=\pi$ and $h=-1$ . The amplitude and period of function is evaluated as shown below.

  $\begin{array}{c}\text{Amplitude}=\left|A\right|\\ =\left|1\right|\left[\text{Put the value of }A\right]\\ =1\end{array}$

The period of function is evaluated as shown below.

  $\begin{array}{c}\text{Period}=\frac{2\pi }{k}\\ =\frac{2\pi }{2}\left[\text{Put the value of }k\right]\\ =\pi \end{array}$

The phase shift of function is evaluated as shown below.

  $\begin{array}{c}\text{Phase shift}=-\frac{c}{k}\\ =-\frac{\pi }{2}\left[\text{Put the value of }c\text{ and }k\right]\end{array}$

Now divide the period into four equal lengths (i.e. $\frac{\pi }{4},\frac{\pi }{2},\frac{3\pi }{4}$ and $\pi$ ) to evaluate the points as shown below.

  $\begin{array}{c}y=\sec \left(2\left(0\right)+\pi \right)-1\left[\text{For }\theta =0\right]\\ =\sec \left(\pi \right)-1\\ =-1-1\\ =-2\end{array}$

  $\begin{array}{c}y=\sec \left(2\left(\frac{\pi }{4}\right)+\pi \right)-1\left[\text{For }\theta =\frac{\pi }{4}\right]\\ =\sec \left(\frac{\pi }{2}\right)-1\\ =undefined\end{array}$

  $\begin{array}{c}y=\sec \left(2\left(\frac{\pi }{2}\right)+\pi \right)-1\left[\text{For }\theta =\frac{\pi }{2}\right]\\ y=\sec \left(2\pi \right)-1\\ =1-1\\ =0\\ y=\sec \left(2\left(\frac{3\pi }{4}\right)+\pi \right)-1\left[\text{For }\theta =\frac{3\pi }{4}\right]\\ y=\sec \left(\frac{3\pi }{2}\right)-1\\ =undefined\\ y=\sec \left(2\left(\pi \right)+\pi \right)-1\left[\text{For }\theta =\pi \right]\\ =\sec \left(3\pi \right)-1\\ =-1-1\\ =-2\end{array}$

Create a table of values as shown below.

$x$	0	$\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$4$}\right.$	$\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$	$\raisebox{1ex}{$3\pi $}\!\left/ \!\raisebox{-1ex}{$4$}\right.$	$\pi$
$y$	$-2$	Undefined	0	undefined	$-2$

Plot the coordinates on the graph and connect them with a smooth curve as shown below.