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1 10.9 Trigonometrie ie GRA RAAD 10_KABV Kurrikulum 1.1 Definieer ieer trigonometriese verhoudings as sinθ, cosθ en tanθ deur reghoekige driehoeke te gebruik. (a (b cosa sinc tana... sina tanc cosc (c (d sinz siny sinq cosz cosy tanr tanz tany cosq 1. Brei die definisies uit vir sinθ, cosθ en tanθ as 0 θ 360. (a cos 100 (b tan 10 (c sin 300 (d tan 135 (e sin 315 (f cos 10 (g sin 40 (h cos 5 (i tan 150 (j sin 135
2 sina cosa tana coseca seca cot A cosec C sec C cotc sin C cos C tan C 1.4 Lei die waardes van die trigonometriese ese verhoudings vir spesiale hoeke af (sonder die gebruik van n sakrekenaar. (a (b (c (d (e (f tan 5.sin135.tan 300 cos 315.cos 5.cos150 tan10 tan 330 sin60.cos30.tan60 sin30.tan45.cos45 tan10.cos 10 sin 40.sin 40 cos 330 cos 5.cos 315.tan Los D probleme op m.b.v. reghoekige driehoeke. (a Die lengte van n mas is 8.5m, en die lengte van die skaduwee van die mas is 7.5m. Bereken die hoogtehoek van die son op daardie oomblik. (b Die dieptehoek van n sweeftuig na n vrou op die grond is 43. As die
3 sweeftuig 340m van die vrou af is, bereken die hoogte van die sweeftuig. (c Twee torings is 1m van mekaar af. Vanaf B is die hoogtehoek na DE, 9 en vanaf D is die hoogtehoek na BC, 48. Bereken die verskil in die hoogte van die twee torings. (d n Gebou (DF en n toring (CE is 94m van mekaar af. Vanaf die dak van die gebou is die hoogtehoek na die top van die toring 15 en die dieptehoek na die voet van die toring is 46. Bereken die hoogte van die toring. 1.6 Los die trigonometriese vergelykings op vir hoeke tussen 0 en 90. (a sin51 cos β (b cos33 sinα
4 (c sin75 cos3θ (d cos 4α sin 5α (e cos ( β 43 sin 65 (f sin(θ + 54 cos (θ Gebruik sketse om die getalwaardes van die verhoudings te bereken vir die hoeke tussen 0 and 360. (a As 17sinA 15, 0 A 90, bereken tan A. (b As 9tan β 40 en β is n skerphoek, bereken sin β. (c As 6sinα 5 0 en α [90 ;180 ], bereken cosα. (d As -5cos β 4 0 en β [180 ;70 ], bereken sin β. (e As 5sinθ 4 0 en 90 θ 180, bereken cosθ.
5 MEMO 1.1 Definieer ieer trigonometriese verhoudings as sinθ, cosθ en tanθ deur reghoekige driehoeke e te gebruik. [ ] (a (b cos A b c sin C b c tan A c a sin A 6 tan C 6 8 cos C 6 (c (d sinz ; siny cosz ; cosy tanz ; tany sinq p q tanr q r cosq p r 1. Brei die definisies uit vir sinθ, cosθ en tanθ as 0 θ 360. [7.4..; ] ]
6 (a cos 100 -cos 80 (b tan 10 tan 30 (c sin 300 -sin 60 (d tan 135 -tan 45 (e sin 315 -sin 45 (f cos 10 - cos 60 (g sin 40 -sin 60 (h cos 5 -cos 45 (i tan 150 -tan 30 (j sin 135 sin Definie ieer r die resiproke van die trigonometriese verhoudings as cosecθ, secθ en cotθ, deur reghoekige driehoeke te gebruik. [7..1.3; ; ; 7..1.] a sin A b c cos A b a tan A c b cosec A a b sec A c c cot A a b cosec C c b sec C a a cot C c c sin C b a cos C b c tan C a 1.4 Lei die waardes van die trigonometriese ese verhoudings vir spesiale hoeke af (sonder die gebruik van n sakrekenaar. [7.3..1; ; ; ] (a
7 tan 5.sin135.tan 300 cos 315.cos 5.cos150 tan 45.sin 45.( tan 60 cos 45.( cos 45.( cos ( (.( (b (c tan10 tan 330 tan 60 tan sin60.cos30.tan (d sin30.tan45.cos (e
8 tan10.cos 10 sin 40.sin 40 ( tan 60.( cos30 ( sin 60.( sin ( (f cos 330 cos 5.cos 315.tan 5 cos30 ( cos 45.cos 45.tan Los D probleme op m.b.v. reghoekige driehoeke. [ ; ; ] (a Die lengte van n mas is 8.5m, en die lengte van die skaduwee van die mas is 7.5m. Bereken die hoogtehoek van die son op daardie oomblik. AB tan BC 8.5 tan 7.5 tan 1 (
9 (b Die dieptehoek van n sweeftuig na n vrou op die grond is 43. As die sweeftuig 340m van die vrou af is, bereken die hoogte van die sweeftuig. AC sin BC sin sin m (c Twee torings is 1m van mekaar af. Vanaf B is die hoogtehoek na DE, 9 en vanaf D is die hoogtehoek na BC, 48. Bereken die verskil in die hoogte van die twee torings. BCD : BC tan 48 BD tan tan m BDE : DE tan 9 BD y tan 9 1 y 1 tan 9 y 6.65m
10 BC DE m (d n Gebou (DF en n toring (CE is 94m van mekaar af. Vanaf die dak van die gebou is die hoogtehoek na die top van die toring 15 en die dieptehoek na die voet van die toring is 46. Bereken die hoogte van die toring. BCD : BC tan 46 DB tan tan m BDE : BE tan15 DB y tan15 94 y 94 tan15 y 5.19m EC m 1.6 Los die trigonometriese vergelykings op vir hoeke tussen 0 en 90. [7.6..1; ; ] (a sin51 cos β sin 51 cos β sin 51 sin(90 β β β β 39 (b cos33 sinα
11 cos33 sinα cos33 cos(90 α α α α 57 (c sin75 cos3θ sin 75 cos3θ sin 75 sin(90 3θ θ 3θ θ 15 θ 5 (d cos 4α sin 5α cos 4α sin 5α cos 4α cos(90 5α 4α 90 5α 9α 90 α 10 (e cos ( β 43 sin 65 cos( β 43 sin 65 cos( β 43 cos(90 65 β β 68 (f sin(θ + 54 cos (θ 8
12 sin( θ + 54 cos( θ 8 sin( θ + 54 sin(90 ( θ 8 sin( θ + 54 sin(90 θ + 8 θ θ + 8 θ θ 44 θ 1.7 Gebruik sketse om die getalwaardes van die verhoudings te bereken vir die hoeke tussen 0 and 360. [ ; ; ; ] (a As 17sinA 15, 0 A 90, bereken tan A. 15 tan A 8 + y + ( r (17 (b As 9tan β 40 en β is n skerphoek, bereken sin β. 40 sin β 41 ( r r + y 1681 r 41 r + (40 r
13 (c As 6sinα 5 0 en α [90 ;180 ], bereken cosα. + y r + (5 ( cosα 11 6 (d As -5cos β 4 0 en β [180 ;70 ], bereken sin β. + y r ( 4 + y ( y y 3 5 sin β 3 5 (e As 5sinθ 4 0 en 90 θ 180, bereken cosθ.
14 + y r + (4 ( cosθ 3 5
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