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Inleiding

In meetkunde leer ons wat die verwantskap tussen die sye en hoeke van veelhoeke is, maar nie hoe om 'n hoek te bereken as ons die lengtes van die sye weet nie. Trigonometrie handel oor die verwantskap tussen die hoeke en die sye van 'n reghoekige driehoek. Ons gaan leer oor trigonometriese funksies (driehoeksmetingfunksies), wat die grondslag van trigonometrie vorm.

Ondersoek: die geskiedenis van trigonometrie

Werk in pare of groepe en ondersoek die geskiedenis van die grondslag van trigonometrie. Beskryf die verskillende stadia van ontwikkeling en hoe die onderstaande kulture trigonometrie gebruik het om hulle lewens te verbeter.

Die werk van die volgende mense of kulture kan ondersoek word:

  1. Kulture
    1. Antieke Egiptenare
    2. Mesopotamiërs
    3. Antieke Indiërs van die Indusvallei
  2. Mense
    1. Lagadha (ongeveer 1350-1200 VC)
    2. Hipparchus (ongeveer 150 BC)
    3. Ptolemy (ongeveer 100)
    4. Aryabhata (ongeveer 499)
    5. Omar Khayyam (1048-1131)
    6. Bhaskara (ongeveer 1150)
    7. Nasir al-Din (13de eeu)
    8. al-Kashi and Ulugh Beg (14de eeu)
    9. Bartholemaeus Pitiscus (1595)

Interessante feit

Jy behoort uit meetkunde bekend te wees met die idee om hoeke te meet, maar het jy al ooit gewonder hoekom daar 360 grade in 'n sirkel is? Die rede is suiwer geskiedkundig. Daar is 360 grade in 'n sirkel omdat die antieke Babiloniërs 'n getallestelsel met grondtal (basis) 60 gehad het. 'n Grondtal is die basisgetal waarby jy nog 'n syfer byvoeg wanneer jy tel. Die getallestelsel wat ons daagliks gebruik word die desimale stelsel genoem (die grondtal is 10), maar rekenaars gebruik die binêre sisteem (die grondtal is 2). 360 = 6 x 60. Dus het dit vir hulle sin gemaak om 360 grade in 'n sirkel te hê.

Die gebruik van trigonometrie

Daar is baie toepassings van trigonometrie. Die tegniek van triangulering, wat in sterrekunde gebruik word om die afstand na nabygeleë sterre te meet, is van besondere waarde in geografie om die afstand tussen landmerke te meet. Satellietnavigasiestelsels soos GPS (globale posisionering stelsel) sou nie moontlik gewees het sonder trigonometrie nie. Ander velde wat gebruik maak van trigonometrie sluit in sterrekunde (en by implikasie navigasie op die oseane, in vliegtuie en in die ruimte), musiek teorie, akoestiek, optika, ontleding van finansiële markte, elektronika, waarskynlikheidsteorie, statistiek, biologie, mediese beeldvorming (CAT-skanderings en ultraklank), farmakologie, chemie, getalleteorie (en dus kriptologie), seismologie, meteorologie, oseanografie, verskeie fisiese wetenskappe, landmeting en geodesie, argitektuur, fonetiek, ekonomie, elektriese ingenieurswese, meganiese ingenieurswese, siviele ingenieurswese, rekenaargrafika, kartografie, kristallografie en spelontwikkeling.

Bespreking: gebruike van trigonometrie

Kies een van die gebruike van trigonometrie uit die gegewe lys en skryf 'n 1-bladsy verslag wat beskryf hoe trigonometrie in jou gekose veld gebruik word.

Gelykvormigheid van driehoeke

As A B C gelykvormig is aan D E F skryf ons dit as volg:

A B C ||| D E F

Dan is dit moontlik om die verhoudings tussen ooreenstemmende sye van die twee driehoeke af te lei:

A B B C = D E E F A B A C = D E D F A C B C = D F E F A B D E = B C E F = A C D F

Die belangrikste feit omtrent gelykvormige driehoeke A B C and D E F is dat die hoek by punt A geyk is aan die hoek by punt D, die hoek by B is gelyk aan die hoek by E, en die hoek by C is gelyk aan die hoek by F.

A = D B = E C = F

Ondersoek: verhoudings tussen gelykvormige driehoeke

In jou oefeningboek, teken drie gelykvormiige driehoeke van verskillende groottes, maar elkeen met A^=30°;Bˆ=90° and Cˆ=60°. Meet die hoeke en lengtes van die driehoeke baie akkuraat om die tabel hieronder te voltooi (rond antwoorde af tot een desimale plek).

Verdeling van die lengtes van sye (Verhoudings)
A B B C = A B A C = C B A C =
A ' B ' B ' C ' = A ' B ' A ' C ' = C ' B ' A ' C ' =
A ' ' B ' ' B ' ' C ' ' = A ' ' B ' ' A ' ' C ' ' = C ' ' B ' ' A ' ' C ' ' =

Watter waarnemings kan jy oor die verhoudings van die sye maak?

Hierdie gelyke verhoudings word gebruik om die trigonometriese funksies te definieer.

Let wel: In algebra gebruik ons dikwels die letter x vir die onbekende veranderlike (alhoewel ons enige ander letter kan gebruik, soos a , b , k , ens). In trigonometrie gebruik ons dikwels die Griekse simbool θ vir 'n onbekende hoek (ons kan ook α , β , γ etc gebruik).

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
hey
Giriraj
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
ya I also want to know the raman spectra
Bhagvanji
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
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Source:  OpenStax, Siyavula textbooks: wiskunde (graad 10) [caps]. OpenStax CNX. Aug 04, 2011 Download for free at http://cnx.org/content/col11328/1.4
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