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Grafieke van trigonometriese funksies

Hierdie afdeling beskryf die grafieke van trigonometriese funksies.

Grafiek van sin θ

Grafiek van sin θ

Volgooi die volgende tabel en gebruik jou sakrekenaar om die waardes te bereken. Stip dan die waardes met sin θ op die y -as en θ op die x -as. Rond die antwoorde af tot 1 desimale plek.

θ 0 30 60 90 120 150
sin θ
θ 180 210 240 270 300 330 360
sin θ

Laat ons terugkyk na ons waardes vir sin θ .

θ 0 30 45 60 90 180
sin θ 0 1 2 1 2 3 2 1 0

Soos jy kan sien, die funksie sin θ het 'n waarde van 0 by θ = 0 . Sy waarde neem egalig toe tot by θ = 90 wanneer sy waarde 1 is. Ons weet ook dat dit later afneem na 0 as θ = 180 . Deur dit alles bymekaar te sit, kan ons 'n idee kry van die volle omvang van die sinuskurwe. Die sinuskurwe word gewys in [link] . Let op die kurwe se vorm, waar elke kurwe die lengte het van 360 . Ons sê die grafiek het 'n periode van 360 . Die hoogte van die kurwe bo (of onder) die x -as word die kurwe se amplitude genoem. Dus is die amplitude van die sinuskurwe is 1.

Die grafiek van y = sin θ

Funksies in die vorm y = a sin ( x ) + q

In die vergelyking, y = a sin ( x ) + q , a en q is konstantes en het verskillende invloede op die grafiek van die funksie. Die algemene vorm van hierdie grafiek word gewys in [link] vir die funksie f ( θ ) = 2 sin θ + 3 .

Grafiek van f ( θ ) = 2 sin θ + 3

Funksies van die vorm y = a sin ( θ ) + q :

  1. Op dieselfde stel asse, trek die volgende grafieke:
    1. a ( θ ) = sin θ - 2
    2. b ( θ ) = sin θ - 1
    3. c ( θ ) = sin θ
    4. d ( θ ) = sin θ + 1
    5. e ( θ ) = sin θ + 2
    Gebruik jou resultate om die invloed van q af te lei.
  2. Op dieselfde stel asse, trek die volgende grafieke:
    1. f ( θ ) = - 2 · sin θ
    2. g ( θ ) = - 1 · sin θ
    3. h ( θ ) = 0 · sin θ
    4. j ( θ ) = 1 · sin θ
    5. k ( θ ) = 2 · sin θ
    Gebruik jou resultate om die invloed van a af te lei.

Dis duidelik dat q 'n vertikale verskuiwing teweegbring. As q = 2 , sal die hele sinusgrafiek 2 eenhede opskuif. As q = - 1 , suif die hele grafiek 1 eenheid af.

Hierdie eienskappe word opgesom in [link] .

Jy behoort te vind dat die waarde van a die hoogte van die pieke van die grafiek beïnvloed. As die grootte van a toeneem, word die pieke hoër. As dit afneem, word die pieke laer.

Tabel wat die algemene vorms en posisies van grafieke en funksies in die vorm y = a sin ( x ) + q opsom
a > 0 a < 0
q > 0
q < 0

Gebied en terrein

Vir f ( θ ) = a sin ( θ ) + q , is die gebied { θ : θ R } omdat daar geen waarde is van θ R waarvoor f ( θ ) ongedefinieerd is nie.

Die terrein van f ( θ ) = a sin θ + q hang daarvan af of die waarde vir a positief of negatief is. Ons sal die twee gevalle afsonderlik oorweeg.

As a > 0 we have:

- 1 sin θ 1 - a a sin θ a ( Vermenigvuldiging met 'n positiewe getal handhaaf die aard van die ongelykheid ) - a + q a sin θ + q a + q - a + q f ( θ ) a + q

Dit vertel ons dat vir alle waardes van θ , f ( θ ) altyd tussen - a + q en a + q is. Daarom as a > 0 , is die terrein van f ( θ ) = a sin θ + q dus { f ( θ ) : f ( θ ) [ - a + q , a + q ] } .

Insgelyks, daar kan getoon word dat as a < 0 , dan is die terrein van f ( θ ) = a sin θ + q is { f ( θ ) : f ( θ ) [ a + q , - a + q ] } . Dit word as 'n oefening gelaat.

Die maklikste manier om die terrein te bepaal is om bloot vir die "bokant" en die "onderkant" van die grafiek te soek.

Snypunte

Die y -snypunt, y i n t , van f ( θ ) = a sin ( x ) + q is eenvoudig die waarde van f ( θ ) by θ = 0 .

y i n t = f ( 0 ) = a sin ( 0 ) + q = a ( 0 ) + q = q

Grafiek van cos θ

Grafiek van cos θ :

Voltooi die volgende tabel, gebruik jou sakrekenaar om die waardes korrek tot 1 desimale plek te bereken. Stip dan die waardes met cos θ op die y -as en θ op die x -as.

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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