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θ 0 30 60 90 120 150
cos θ
θ 180 210 240 270 300 330 360
cos θ

Let us look back at our values for cos θ

θ 0 30 45 60 90 180
cos θ 1 3 2 1 2 1 2 0 - 1

If you look carefully, you will notice that the cosine of an angle θ is the same as the sine of the angle 90 - θ . Take for example,

cos 60 = 1 2 = sin 30 = sin ( 90 - 60 )

This tells us that in order to create the cosine graph, all we need to do is to shift the sine graph 90 to the left. The graph of cos θ 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.

The graph of cos θ .

Functions of the form y = a cos ( x ) + q

In the equation, y = a cos ( x ) + 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 ( θ ) = 2 cos θ + 3 .

Graph of f ( θ ) = 2 cos θ + 3

Functions of the form y = a cos ( θ ) + q :

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

Table summarising general shapes and positions of graphs of functions of the form y = a cos ( x ) + q .
a > 0 a < 0
q > 0
q < 0

Domain and range

For f ( θ ) = a cos ( θ ) + q , the domain is { θ : θ R } because there is no value of θ R for which f ( θ ) is undefined.

It is easy to see that the range of f ( θ ) will be the same as the range of a sin ( θ ) + q . This is because the maximum and minimum values of a cos ( θ ) + q will be the same as the maximum and minimum values of a sin ( θ ) + q .

Intercepts

The y -intercept of f ( θ ) = a cos ( x ) + q is calculated in the same way as for sine.

y i n t = f ( 0 ) = a cos ( 0 ) + q = a ( 1 ) + q = a + q

Comparison of graphs of sin θ And cos θ

The graph of cos θ (solid-line) and the graph of sin θ (dashed-line).

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 . That means that if you shift the whole cos graph to the right by 90 it will overlap perfectly with the sin graph. You could also move the sin graph by 90 to the left and it would overlap perfectly with the cos graph. This means that:

sin θ = cos ( θ - 90 ) ( shift the cos graph to the right ) a nd cos θ = sin ( θ + 90 ) ( shift the sin graph to the left )

Graph of tan θ

Graph of tan θ

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

θ 0 30 60 90 120 150
tan θ
θ 180 210 240 270 300 330 360
tan θ

Questions & Answers

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
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
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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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