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Graphs of trigonometric functions

This section describes the graphs of trigonometric functions.

Graph of sin θ

Graph of sin θ

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

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

Let us look back at our values for sin θ

θ 0 30 45 60 90 180
sin θ 0 1 2 1 2 3 2 1 0

As you can see, the function sin θ has a value of 0 at θ = 0 . Its value then smoothly increases until θ = 90 when its value is 1. We also know that it later decreases to 0 when θ = 180 . Putting all this together we can start to picture the full extent of the sine graph. The sine graph is shown in [link] . Notice the wave shape, with each wave having a length of 360 . We say the graph has a period of 360 . The height of the wave above (or below) the x -axis is called the wave's amplitude . Thus the maximum amplitude of the sine-wave is 1, and its minimum amplitude is -1.

The graph of sin θ .

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

In the equation, y = a sin ( 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 sin θ + 3 .

Graph of f ( θ ) = 2 sin θ + 3

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

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

You should have found that the value of a affects the height of the peaks of the graph. As the magnitude of a increases, the peaks get higher. As it decreases, the peaks get lower.

q is called the vertical shift . If q = 2 , then the whole sine graph shifts up 2 units. If q = - 1 , the whole sine graph shifts down 1 unit.

These different properties are summarised in [link] .

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

Domain and range

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

The range of f ( θ ) = a sin θ + q depends on whether the value for a is positive or negative. We will consider these two cases separately.

If a > 0 we have:

- 1 sin θ 1 - a a sin θ a ( Multiplication by a positive number maintains the nature of the inequality ) - a + q a sin θ + q a + q - a + q f ( θ ) a + q

This tells us that for all values of θ , f ( θ ) is always between - a + q and a + q . Therefore if a > 0 , the range of f ( θ ) = a sin θ + q is { f ( θ ) : f ( θ ) [ - a + q , a + q ] } .

Similarly, it can be shown that if a < 0 , the range of f ( θ ) = a sin θ + q is { f ( θ ) : f ( θ ) [ a + q , - a + q ] } . This is left as an exercise.

The easiest way to find the range is simply to look for the "bottom" and the "top" of the graph.

Intercepts

The y -intercept, y i n t , of f ( θ ) = a sin ( x ) + q is simply the value of f ( θ ) at θ = 0 .

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

Graph of cos θ

Graph of cos θ :

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

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
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
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
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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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