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

 Page 2 / 12
$\begin{array}{ccc}\hfill y& =& cos\left(k\theta \right)\hfill \\ \hfill {y}_{int}& =& cos\left(0\right)\hfill \\ & =& 1\hfill \end{array}$

## 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 $f\left(\theta \right)=cos\left(\theta +{30}^{\circ }\right)$ .

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
Got questions? Join the online conversation and get instant answers! By Tamsin Knox By Jugnu Khan By OpenStax By Brooke Delaney By Brooke Delaney By By Jessica Collett By Saylor Foundation By Madison Christian By JavaChamp Team