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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

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Shanjida
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
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I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
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Porter
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Cesar
I'm interested in nanotube
Uday
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AMJAD
preparation of nanomaterial
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
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AMJAD
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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
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Damian
silver nanoparticles could handle the job?
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Azam
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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
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Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele 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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