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Die trigonometriese funksies vir enige hoek

Tot dusver het ons die trigonometriese funksies gedefinieer deur gebruik te maak van reghoekige driehoeke. Ons kan nou hierdie definisies uitbrei na alle hoeke. Ons kry dit reg deur daarop te let dat die definisies nie afhanklik is van die lengtes van die sye van die driehoek nie, maar slegs bepaal word deur die hoekgootte. So, as ons enige punt op die Cartesiese vlak merk en 'n lyn trek vanaf daardie punt na die oorsprong, kan ons werk met die hoek tussen daardie lyn en die x-as. In [link] is punte P en Q gemerk. 'n Lyn is getrek vanaf die oorsprong na elk van die punte. Die stippellyne toon hoe ons reghoekige driehoeke kan konstureer vir elke punt. Nou kan ons hoeke A en B vind.

Jy sal vind hoek A is 63 , 43 . Vir hoek B, moet jy eers vir x = 33 , 69 bereken en dan is B = 180 - 33 , 69 = 146 , 31 . Maar, gestel ons dit wil doen sonder om hierdie hoeke uit te werk en vas te stel of ons 180 grade of 90 grade moet bytel of aftrek? Kan ons trigonometriese funksies gebruik om dit te doen? Beskou punt P in [link] . Om die hoek te vind, sou jy een van die trigonometriese funksies gebruik het, naamlik tan θ . Let op, die sy wat aangrensend is aan die hoek, is die x-koördinaat en die sy teenoor die hoek is die y-koördinaat. Maar wat van die skuinssy? Ons kan dit vind deur die Stelling van Pythagoras te gebruik aangesien ons die twee reghoeksye van 'n reghoekige driehoek het. As ons 'n sirkel trek met die oorsprong as middelpunt, dan is die lengte vanaf die oorsprong na punt P die radius van die sirkel, wat ons aandui met r. Nou kan ons al ons trigonometriese verhoudings herskryf in terme van x, y en r. Maar hoe help dit ons om B te kry? Vanaf punt Q na die oorsprong is r en ons het die koördinate van Q. Ons gebruik nou eenvoudig ons nuut-gedefinieërde trigonometriese funksies om B te bereken! (Probeer dit self en bevestig dat jy dieselfde antwoord kry as vantevore). Wanneer ons anti-kloksgewys om die oorsprong beweeg, is die hoeke positief en wanneer ons kloksgewys draai in die Cartesiese vlak, is die hoeke negatief.

Ons kry dus die volgende definisies vir die trigonometriese funksies:

sin θ = x r cos θ = y r tan θ = y x

Gestel die x-koördinaat of die y-koördinaat is negatief. Ignoreer ons dit, of is daar 'n manier om dit in berekening te bring? Die antwoord is dat ons dit nie ignoreer nie: Die teken voor die x- of y-koördinaat bepaal of sin, cos en tan positief of negatief is. Die Cartesiese vlak is verdeel in kwadrante en ons gebruik dan [link] om vir ons aan te dui of die trigonometriese funksie positief of negatief is. Die diagram staan bekend as die CAST diagram.

Op dieselfde wyse kan ons die definisies uitbrei na die resiprookfunksies:

cosec θ = r x sec θ = r y cot θ = x y

Punt R(-1;-3) en punt S(3;-3) is aangedui op die diagram hieronder. Vind die hoeke α en β .

  1. Ons het die koördinate van punte R en S en ons moet die groottes van die twee hoeke vind. Hoek β is positief en hoek α is negatief.

  2. Ons gebruik tan om β te vind, aangesien ons slegs x en y het. Ons sien die hoek lê in die derde kwadrant, waar tan positief is.

    tan ( β ) = y x tan ( β ) = - 3 - 1 β = tan - 1 ( 3 ) β = 71 , 57
  3. Ons gebruik tan om α te bereken aangesien ons x en y het. Die hoek is in die vierde kwadrant, waar tan negatief is.

    tan ( α ) = y x tan ( α ) = - 3 3 α = tan - 1 ( - 1 ) α = - 45
  4. Hoek α is - 45 en hoek β is 71 , 57

Questions & Answers

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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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