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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 .


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

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
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..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
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.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit 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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