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Trigonometry - grade 11

History of trigonometry

Work in pairs or groups and investigate the history of the development of trigonometry. Describe the various stages of development and how different cultures used trigonometry to improve their lives.

The works of the following people or cultures can be investigated:

  1. Cultures
    1. Ancient Egyptians
    2. Mesopotamians
    3. Ancient Indians of the Indus Valley
  2. People
    1. Lagadha (circa 1350-1200 BC)
    2. Hipparchus (circa 150 BC)
    3. Ptolemy (circa 100)
    4. Aryabhata (circa 499)
    5. Omar Khayyam (1048-1131)
    6. Bhaskara (circa 1150)
    7. Nasir al-Din (13th century)
    8. al-Kashi and Ulugh Beg (14th century)
    9. Bartholemaeus Pitiscus (1595)

Graphs of trigonometric functions

Functions of the form y = sin ( k θ )

In the equation, y = sin ( k θ ) , 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 ( θ ) = sin ( 2 θ ) .

Graph of f ( θ ) = sin ( 2 θ ) (solid line) and the graph of g ( θ ) = sin ( θ ) (dotted line).

Functions of the form y = sin ( k θ )

On the same set of axes, plot the following graphs:

  1. a ( θ ) = sin 0 , 5 θ
  2. b ( θ ) = sin 1 θ
  3. c ( θ ) = sin 1 , 5 θ
  4. d ( θ ) = sin 2 θ
  5. e ( θ ) = sin 2 , 5 θ

Use your results to deduce the effect of k .

You should have found that the value of k affects the period or frequency of the graph. Notice that in the case of the sine graph, the period (length of one wave) is given by 360 k .

These different properties are summarised in [link] .

Table summarising general shapes and positions of graphs of functions of the form y = sin ( k x ) . The curve y = sin ( x ) is shown as a dotted line.
k > 0 k < 0

Domain and range

For f ( θ ) = sin ( k θ ) , 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 ) = sin ( k θ ) is { f ( θ ) : f ( θ ) [ - 1 , 1 ] } .

Intercepts

For functions of the form, y = sin ( k θ ) , the details of calculating the intercepts with the y axis are given.

There are many x -intercepts.

The y -intercept is calculated by setting θ = 0 :

y = sin ( k θ ) y i n t = sin ( 0 ) = 0

Functions of the form y = cos ( k θ )

In the equation, y = cos ( k θ ) , 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 ( θ ) = cos ( 2 θ ) .

Graph of f ( θ ) = cos ( 2 θ ) (solid line) and the graph of g ( θ ) = c o s ( θ ) (dotted line).

Functions of the form y = cos ( k θ )

On the same set of axes, plot the following graphs:

  1. a ( θ ) = cos 0 , 5 θ
  2. b ( θ ) = cos 1 θ
  3. c ( θ ) = cos 1 , 5 θ
  4. d ( θ ) = cos 2 θ
  5. e ( θ ) = cos 2 , 5 θ

Use your results to deduce the effect of k .

You should have found that the value of k affects the period or frequency of the graph. The period of the cosine graph is given by 360 k .

These different properties are summarised in [link] .

Table summarising general shapes and positions of graphs of functions of the form y = cos ( k x ) . The curve y = cos ( x ) is plotted with a dotted line.
k > 0 k < 0

Domain and range

For f ( θ ) = cos ( k θ ) , 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 ) = cos ( k θ ) is { f ( θ ) : f ( θ ) [ - 1 , 1 ] } .

Intercepts

For functions of the form, y = cos ( k θ ) , the details of calculating the intercepts with the y axis are given.

The y -intercept is calculated as follows:

Questions & Answers

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
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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
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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
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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