# Definitions, simple applications, and graphs of trigonometric  (Page 6/7)

 Page 6 / 7
 $\theta$ 0 ${}^{\circ }$ 30 ${}^{\circ }$ 60 ${}^{\circ }$ 90 ${}^{\circ }$ 120 ${}^{\circ }$ 150 ${}^{\circ }$ $cos\theta$ $\theta$ 180 ${}^{\circ }$ 210 ${}^{\circ }$ 240 ${}^{\circ }$ 270 ${}^{\circ }$ 300 ${}^{\circ }$ 330 ${}^{\circ }$ 360 ${}^{\circ }$ $cos\theta$

Let us look back at our values for $cos\theta$

 $\theta$ ${0}^{\circ }$ ${30}^{\circ }$ ${45}^{\circ }$ ${60}^{\circ }$ ${90}^{\circ }$ ${180}^{\circ }$ $cos\theta$ 1 $\frac{\sqrt{3}}{2}$ $\frac{1}{\sqrt{2}}$ $\frac{1}{2}$ 0 $-1$

If you look carefully, you will notice that the cosine of an angle $\theta$ is the same as the sine of the angle ${90}^{\circ }-\theta$ . Take for example,

$cos{60}^{\circ }=\frac{1}{2}=sin{30}^{\circ }=sin\left({90}^{\circ }-{60}^{\circ }\right)$

This tells us that in order to create the cosine graph, all we need to do is to shift the sine graph ${90}^{\circ }$ to the left. The graph of $cos\theta$ is shown in [link] . As the cosine graph is simply a shifted sine graph, it will have the same period and amplitude as the sine graph.

## Functions of the form $y=acos\left(x\right)+q$

In the equation, $y=acos\left(x\right)+q$ , $a$ and $q$ are constants and have 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 $f\left(\theta \right)=2cos\theta +3$ .

## Functions of the form $y=acos\left(\theta \right)+q$ :

1. On the same set of axes, plot the following graphs:
1. $a\left(\theta \right)=cos\theta -2$
2. $b\left(\theta \right)=cos\theta -1$
3. $c\left(\theta \right)=cos\theta$
4. $d\left(\theta \right)=cos\theta +1$
5. $e\left(\theta \right)=cos\theta +2$
Use your results to deduce the effect of $q$ .
2. On the same set of axes, plot the following graphs:
1. $f\left(\theta \right)=-2·cos\theta$
2. $g\left(\theta \right)=-1·cos\theta$
3. $h\left(\theta \right)=0·cos\theta$
4. $j\left(\theta \right)=1·cos\theta$
5. $k\left(\theta \right)=2·cos\theta$
Use your results to deduce the effect of $a$ .

You should have found that the value of $a$ affects the amplitude of the cosine graph in the same way it did for the sine graph.

You should have also found that the value of $q$ shifts the cosine graph in the same way as it did the sine graph.

These different properties are summarised in [link] .

 $a>0$ $a<0$ $q>0$ $q<0$

## Domain and range

For $f\left(\theta \right)=acos\left(\theta \right)+q$ , 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.

It is easy to see that the range of $f\left(\theta \right)$ will be the same as the range of $asin\left(\theta \right)+q$ . This is because the maximum and minimum values of $acos\left(\theta \right)+q$ will be the same as the maximum and minimum values of $asin\left(\theta \right)+q$ .

## Intercepts

The $y$ -intercept of $f\left(\theta \right)=acos\left(x\right)+q$ is calculated in the same way as for sine.

$\begin{array}{ccc}\hfill {y}_{int}& =& f\left({0}^{\circ }\right)\hfill \\ & =& acos\left({0}^{\circ }\right)+q\hfill \\ & =& a\left(1\right)+q\hfill \\ & =& a+q\hfill \end{array}$

## Comparison of graphs of $sin\theta$ And $cos\theta$

Notice that the two graphs look very similar. Both oscillate up and down around the $x$ -axis as you move along the axis. The distances between the peaks of the two graphs is the same and is constant along each graph. The height of the peaks and the depths of the troughs are the same.

The only difference is that the $sin$ graph is shifted a little to the right of the $cos$ graph by 90 ${}^{\circ }$ . That means that if you shift the whole $cos$ graph to the right by 90 ${}^{\circ }$ it will overlap perfectly with the $sin$ graph. You could also move the $sin$ graph by 90 ${}^{\circ }$ to the left and it would overlap perfectly with the $cos$ graph. This means that:

$\begin{array}{ccc}\hfill sin\theta & =& cos\left(\theta -90\right)\phantom{\rule{1.em}{0ex}}\left(\mathrm{shift}\phantom{\rule{2.pt}{0ex}}\mathrm{the}cos\mathrm{graph}\phantom{\rule{2.pt}{0ex}}\mathrm{to}\phantom{\rule{2.pt}{0ex}}\mathrm{the}\phantom{\rule{2.pt}{0ex}}\mathrm{right}\right)\hfill \\ & \mathbf{a}\mathrm{nd}& \\ \hfill cos\theta & =& sin\left(\theta +90\right)\phantom{\rule{1.em}{0ex}}\left(\mathrm{shift}\phantom{\rule{2.pt}{0ex}}\mathrm{the}sin\mathrm{graph}\phantom{\rule{2.pt}{0ex}}\mathrm{to}\phantom{\rule{2.pt}{0ex}}\mathrm{the}\phantom{\rule{2.pt}{0ex}}\mathrm{left}\right)\hfill \end{array}$

## Graph of $tan\theta$

Complete the following table, using your calculator to calculate the values correct to 1 decimal place. Then plot the values with $tan\theta$ on the $y$ -axis and $\theta$ on the $x$ -axis.

 $\theta$ 0 ${}^{\circ }$ 30 ${}^{\circ }$ 60 ${}^{\circ }$ 90 ${}^{\circ }$ 120 ${}^{\circ }$ 150 ${}^{\circ }$ $tan\theta$ $\theta$ 180 ${}^{\circ }$ 210 ${}^{\circ }$ 240 ${}^{\circ }$ 270 ${}^{\circ }$ 300 ${}^{\circ }$ 330 ${}^{\circ }$ 360 ${}^{\circ }$ $tan\theta$

#### Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
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
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

### Read also:

#### Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Math 1508 (lecture) readings in precalculus. OpenStax CNX. Aug 24, 2011 Download for free at http://cnx.org/content/col11354/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Math 1508 (lecture) readings in precalculus' conversation and receive update notifications?

 By OpenStax By OpenStax By Brooke Delaney By Brooke Delaney By Richley Crapo By OpenStax By OpenStax By OpenStax By OpenStax By Mackenzie Wilcox