# 11.1 Graphs, trigonometric identities, and solving trigonometric  (Page 2/12)

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$\begin{array}{ccc}\hfill y& =& cos\left(k\theta \right)\hfill \\ \hfill {y}_{int}& =& cos\left(0\right)\hfill \\ & =& 1\hfill \end{array}$

## Functions of the form $y=tan\left(k\theta \right)$

In the equation, $y=tan\left(k\theta \right)$ , $k$ is a constant and has different effects on the graph of the function. The general shape of the graph of functions of this form is shown in [link] for the function $f\left(\theta \right)=tan\left(2\theta \right)$ . The graph of tan ( 2 θ ) (solid line) and the graph of g ( θ ) = tan ( θ ) (dotted line). The asymptotes are shown as dashed lines.

## Functions of the form $y=tan\left(k\theta \right)$

On the same set of axes, plot the following graphs:

1. $a\left(\theta \right)=tan0,5\theta$
2. $b\left(\theta \right)=tan1\theta$
3. $c\left(\theta \right)=tan1,5\theta$
4. $d\left(\theta \right)=tan2\theta$
5. $e\left(\theta \right)=tan2,5\theta$

Use your results to deduce the effect of $k$ .

You should have found that, once again, the value of $k$ affects the periodicity (i.e. frequency) of the graph. As $k$ increases, the graph is more tightly packed. As $k$ decreases, the graph is more spread out. The period of the tan graph is given by $\frac{{180}^{\circ }}{k}$ .

These different properties are summarised in [link] .

 $k>0$ $k<0$  ## Domain and range

For $f\left(\theta \right)=tan\left(k\theta \right)$ , the domain of one branch is $\left\{\theta :\theta \in \left(-\frac{{90}^{\circ }}{k},\frac{{90}^{\circ }}{k}\right)\right\}$ because the function is undefined for $\theta =-\frac{{90}^{\circ }}{k}$ and $\theta =\frac{{90}^{\circ }}{k}$ .

The range of $f\left(\theta \right)=tan\left(k\theta \right)$ is $\left\{f\left(\theta \right):f\left(\theta \right)\in \left(-\infty ,\infty \right)\right\}$ .

## Intercepts

For functions of the form, $y=tan\left(k\theta \right)$ , the details of calculating the intercepts with the $x$ and $y$ axis are given.

There are many $x$ -intercepts; each one is halfway between the asymptotes.

The $y$ -intercept is calculated as follows:

$\begin{array}{ccc}\hfill y& =& tan\left(k\theta \right)\hfill \\ \hfill {y}_{int}& =& tan\left(0\right)\hfill \\ & =& 0\hfill \end{array}$

## Asymptotes

The graph of $tank\theta$ has asymptotes because as $k\theta$ approaches ${90}^{\circ }$ , $tank\theta$ approaches infinity. In other words, there is no defined value of the function at the asymptote values.

## Functions of the form $y=sin\left(\theta +p\right)$

In the equation, $y=sin\left(\theta +p\right)$ , $p$ is a constant and has different effects on the graph of the function. The general shape of the graph of functions of this form is shown in [link] for the function $f\left(\theta \right)=sin\left(\theta +{30}^{\circ }\right)$ . Graph of f ( θ ) = sin ( θ + 30 ∘ ) (solid line) and the graph of g ( θ ) = sin ( θ ) (dotted line).

## Functions of the form $y=sin\left(\theta +p\right)$

On the same set of axes, plot the following graphs:

1. $a\left(\theta \right)=sin\left(\theta -{90}^{\circ }\right)$
2. $b\left(\theta \right)=sin\left(\theta -{60}^{\circ }\right)$
3. $c\left(\theta \right)=sin\theta$
4. $d\left(\theta \right)=sin\left(\theta +{90}^{\circ }\right)$
5. $e\left(\theta \right)=sin\left(\theta +{180}^{\circ }\right)$

Use your results to deduce the effect of $p$ .

You should have found that the value of $p$ affects the position of the graph along the $y$ -axis (i.e. the $y$ -intercept) and the position of the graph along the $x$ -axis (i.e. the phase shift ). The $p$ value shifts the graph horizontally. If $p$ is positive, the graph shifts left and if $p$ is negative tha graph shifts right.

These different properties are summarised in [link] .

 $p>0$ $p<0$  ## Domain and range

For $f\left(\theta \right)=sin\left(\theta +p\right)$ , the domain is $\left\{\theta :\theta \in \mathbb{R}\right\}$ because there is no value of $\theta \in \mathbb{R}$ for which $f\left(\theta \right)$ is undefined.

The range of $f\left(\theta \right)=sin\left(\theta +p\right)$ is $\left\{f\left(\theta \right):f\left(\theta \right)\in \left[-1,1\right]\right\}$ .

## Intercepts

For functions of the form, $y=sin\left(\theta +p\right)$ , the details of calculating the intercept with the $y$ axis are given.

The $y$ -intercept is calculated as follows: set $\theta ={0}^{\circ }$

$\begin{array}{ccc}\hfill y& =& sin\left(\theta +p\right)\hfill \\ \hfill {y}_{int}& =& sin\left(0+p\right)\hfill \\ & =& sin\left(p\right)\hfill \end{array}$

## Functions of the form $y=cos\left(\theta +p\right)$

In the equation, $y=cos\left(\theta +p\right)$ , $p$ is a constant and has different effects on the graph of the function. The general shape of the graph of functions of this form is shown in [link] for the function $f\left(\theta \right)=cos\left(\theta +{30}^{\circ }\right)$ .

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Source:  OpenStax, Math 1508 (lecture) readings in precalculus. OpenStax CNX. Aug 24, 2011 Download for free at http://cnx.org/content/col11354/1.1
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