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Inleiding en kernbegrippe

Module by: Free High School Science Texts Project. E-mail the author

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)

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

▵ A B C ||| ▵ D E F ▵ A B C ||| ▵ D E F
(1)

Figure 1
Figure 1 (MG10C15_001.png)

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 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
(2)

Die belangrikste feit omtrent gelykvormige driehoeke ABCABC and DEFDEF 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 ∠ A = ∠ D ∠ B = ∠ E ∠ C = ∠ F
(3)

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).

Figure 2
Figure 2 (MG10C15_002.png)

Table 1
Verdeling van die lengtes van sye (Verhoudings)
A B B C = A B B C = A B A C = A B A C = C B A C = C B A C =
A ' B ' B ' C ' = A ' B ' B ' C ' = A ' B ' A ' C ' = A ' B ' A ' C ' = C ' B ' A ' C ' = C ' B ' A ' C ' =
A ' ' B ' ' B ' ' C ' ' = A ' ' B ' ' B ' ' C ' ' = A ' ' B ' ' A ' ' C ' ' = A ' ' B ' ' A ' ' C ' ' = C ' ' 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 xx vir die onbekende veranderlike (alhoewel ons enige ander letter kan gebruik, soos aa, bb, kk, ens). In trigonometrie gebruik ons dikwels die Griekse simbool θθ vir 'n onbekende hoek (ons kan ook αα , ββ , γγ etc gebruik).

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