Chapter 6.7, Problem 8CFU

To determine

To graph: the trigonometric function.

Explanation of Solution

Given information:

  $y=\tan \left(\theta +\frac{\pi }{4}\right)$

Calculation:

The given function is in the form of $y=A\tan kx+c$ . Compare the given function with the cosine function to get $x=\theta ,A=1,k=1$ and $c=\frac{\pi }{4}$ . 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.

  $\begin{array}{c}\text{Period}=\frac{\pi }{k}\\ =\frac{\pi }{1}\left[\text{Put the value of }k\right]\\ =\pi \end{array}$

The phase shift of function is evaluated as shown below.

  $\begin{array}{c}\text{Phase shift}=\frac{c}{k}\\ =\frac{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{1}\left[\text{Put the value of }c\text{ and }k\right]\\ =\frac{\pi }{4}\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.

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

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

  $\begin{array}{c}y=\tan \left(\frac{\pi }{2}+\frac{\pi }{4}\right)\left[\text{For }\theta =\frac{\pi }{2}\right]\\ =\tan \left(\frac{3\pi }{4}\right)\\ =-1\\ y=\tan \left(\frac{3\pi }{4}+\frac{\pi }{4}\right)\left[\text{For }\theta =\frac{3\pi }{4}\right]\\ =\tan \left(\pi \right)\\ =0\\ y=\tan \left(\pi +\frac{\pi }{4}\right)\left[\text{For }\theta =\pi \right]\\ =\tan \left(\frac{5\pi }{4}\right)\\ =1\end{array}$

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.$	$\pi$
$y$	1	Undefined	$-1$	0	1

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