<< Chapter < Page Chapter >> Page >
Let op dat in die uitgewerkte voorbeeld hierbo, hoek α eenvoudig die hoek is wat lyn OS maak met die x-as. Dus kan ons trigonometrie gebruik om te bereken watter hoek 'n lyn maak met die x- of y-as.

Die oplossing van eenvoudige trigonometriese vergelykings

Deur te gebruik wat ons geleer het omtrent trigonometriese funksies, kan ons nou eenvoudige trigonometriese vergelykings oplos. Ons gebruik ook die beginsels van Equations and Inequalities om ons te help om trigonometriese vergelykings op te los.

Die is belangrik om daarop te let dat 2 sin θ sin ( 2 θ ) . Met ander woorde, om die verhouding te verdubbel (met 2 te vermenigvuldig) het 'n ander betekenis as om die hoek te verdubbel.

Los die volgende trigonometriese vergeyking op: 3 cos ( 2 x + 38 ) + 3 = 2

  1. 3 cos ( 2 x + 38 ) = 2 - 3 cos ( 2 x + 38 ) = - 1 3 ( 2 x + 38 ) = 107 , 46 2 x = 107 , 46 - 38 2 x = 69 , 46 x = 34 , 73
  2. x = 34 , 73
In grade 11 en 12, sal jy meer leer oor die oplos van trigonometriese vergelykings.

Eenvoudige toepassings van trigonometriese funksies

Trigonometrie is waarskynlik in antieke beskawings uitgevind om praktiese probleme, byvoorbeeld in die bou- en konstruksiebedryf, asook navigasie met behulp van sterre, op te los. In hierdie afdeling sal ons wys hoe trigonometrie gebruik kan word om 'n paar ander praktiese probleme op te los.

Hoogte en diepte

Bepaling van die hoogte van 'n gebou deur trigonometrie te gebruik

'n Eenvoudige taak is om die hoogte van 'n gebou te vind met behulp van trigonometrie. Ons sou net 'n maatband van die dak kon laat sak, maar dit is onprakties (en gevaarlik) by hoë geboue. Dit is baie meer sinvol om 'n afstand op die grond te meet en trigonometrie te gebruik om die hoogte van die gebou te vind.

[link] toon 'n gebou waarvan ons nie die hoogte weet nie. Ons het 100 m weg van die gebou gestap en die hoek van die grond tot by die top van die gebou gemeet . Hierdie hoek is 38 , 7 . Ons noem hierdie hoek die hoogtehoek . Soos jy kan sien van [link] , het ons nou 'n reghoekige driehoek. Omdat ons weet wat die lengte van een sy en 'n hoek is, kan ons die hoogte van die driehoek bereken, wat die hoogte van die gebou is wat ons probeer vind.

As ons kyk na die figuur, sien ons dat ons met die teenoorstaande en die aangrensende sy van die hoogtehoek werk en ons kan skryf:

tan 38 , 7 = teenoorstaande aangrensend = hoogte 100 m hoogte = 100 m × tan 38 , 7 = 80 m

'n Blok woonstelle is 100m weg van 'n selfoontoring. Iemand staan by B . Hulle meet die hoek van B na die bopunt van die toring E en dit is 62 . Dit is die hoogtehoek. Dan meet hulle die hoek van B af na die basis van die toring C en dit is 34 . Dit is die dieptehoek. Wat is die hoogte van die selfoontoring korrek tot 1 desimale plek?

  1. Om die hoogte van 'n toring te vind, hoef ons net die lengte van C D en D E te vind. Ons sien dat B D E en B D C beide reghoekige driehoeke is. Vir elkeen van die driehoeke het ons 'n hoek en ons het die lengte B D . Dus kan ons die sye van die driehoeke bereken.

  2. Dit word vir ons gegee dat die lengte van A C 100m is. C A B D is 'n reghoek, dus B D = A C = 100 m .

    tan ( C B ^ D ) = C D B D C D = B D × tan ( C B ^ D ) = 100 × tan 34

    Gebruik jou sakrekenaar om te vind dat tan 34 = 0 , 6745 . Deur dit te gebruik, vind ons dat C D = 67 , 45 m.

  3. tan ( D B ^ E ) = D E B D D E = B D × tan ( D B ^ E ) = 100 × tan 62 = 188 , 07 m
  4. Ons het die hoogte van die toring C E = C D + D E = 67 , 45 m + 188 , 07 m = 255 . 5 m .

Kaarte en planne

Kaarte en planne is gewoonlik skaaltekeninge. Dit beteken hulle is 'n presiese kopie van die regte ding, maar gewoonlik kleiner. Dus word net lengtes verander, maar al die hoeke is dieselfde. Ons kan dus hierdie idee gebruik om kaarte en planne te gebruik deur inligting van die werklike wêreld by te voeg.

'n Skip op pad na die Kaapstadhawe bereik punt A op die kaart, reg suid van Pretoria en reg oos van Kaapstad. As die afstand vanaf Kaapstad na Pretoria 1000km is, gebruik trigonometrie om uit te vind hoe ver oos die skip van Kaapstad is, en vind op hierdie manier die skaal van die kaart.

  1. Ons weet reeds die afstand tussen Kaapstad en A in blokke van die gegewe kaart, is 5 blokke. Dus, as ons bereken hoeveel kilometers hierdie afstand is, kan ons bereken hoeveel kilometers elke blok verteenwoordig, en dan het ons die skaal van die kaart.

  2. Laat ons Kaapstad aandui met C en Pretoria met P . Ons kan sien dat die driehoek A P C reghoekig is. Verder sien ons A C en afstand A P is beide 5 blokke. Dit is dus 'n gelykbenige driehoek en A C ^ P = A P ^ C = 45 .

  3. C A = C P × cos ( A C ^ P ) = 1000 × cos ( 45 ) = 1000 2 km

    Om die skaal uit te werk, sien ons dat

    5 blokke = 1000 2 km 1 blok = 200 2 km

Mnr Nkosi het 'n motorhuis by sy huis, en hy besluit hy wil 'n sinkdak aan die kant van sy motorhuis aanlas. Die motorhuis is 4m hoog, en die plaat vir die dak is 5m lank. As hy die dak teen 'n hoek van 5 wil hê, hoe hoog moet hy die muur, B D , wat die dak ophou, bou? Gee die antwoord tot 2 desimale plekke.

  1. Ons sien dat die driehoek A B C 'n reghoekige driehoek is. Aangesien ons een sy en 'n hoek van die driehoek het, kan ons A C bereken. Die hoogte van die muur is die hoogte van die motorhuis minus A C .

  2. As B C =5m, en hoek A B ^ C = 5 , dan

    A C = B C × sin ( A B ^ C ) = 5 × sin 5 = 5 × 0 , 0871 = 0 . 4358 m

    Dus het ons dat die hoogte van die muur B D = 4 m - 0 . 4358 m = 3 . 56 m .

Toepassings van trigonometriese funksies

  1. 'n Seun vlieg 'n vlieër en staan 30 m van 'n punt direk onder die vlieër. As die tou van die vlieër 50 m lank is, bepaal die hoogtehoek van die vlieër.
  2. Wat is die hoogtehoek van die son as 'n boom van 7,15 m hoog 'n skadu van 10,1 m lank gooi?

Questions & Answers

how environment affect demand and supply of commodity ?
Amos Reply
Wht at the criteria for market ?
what is difference between monitory policy and fiscal policy?
Malik Reply
monetary policy is a policy thrust by National Govt(CBN) to influence government spending, purchase &taxes
necessity of economics
Pamela Reply
I will say want,choice,opportunity cost,scarcity,scale of preference
what is monopoly market.How price output are determined under monopoly market
b) Monopoly market is an impecfect market where s single firm having the innovation to produce a particular commodity.Prices are determined through output since there are no other competitive.
Monopoly market:firm has market power & does not respond to market price
Explain the process of price determination under perfect competition market with suitable diagram
bisham Reply
Price determination under perfect competition via this process :firms have no market power to influence price rather firms respond to market price.
price is different from demand- demand is amount of commodity
Effah Reply
demand is amount /quantity of commodity a potential buyer is willing to buy at a given price at market
demand is a desire of customer on commodity with the ability to pay it and willing to buy it at given price of commodity
demand is price of what
Faith Reply
show that shortrun average cost
Baby Reply
what is economics
Mbah Reply
what is money
what is money
Difine macro economics
money is a medium of exchange between goods and services,maybe inform of currency.
Economics is study of how human beings strive to satisfy numerous wants using limited available resources.
how do you find the maximum number of workers the firms should employ order to produce where there are increasing returns
what are implications of computing national income?.
what is the formulae for calculating national income
it calculated by value added method
classify the production units like agriculture, banking, transport etc
money is anything that is generally acceptetable for human
Estimate the net value added(NVA) at fixed cost by each industrial structure
definition of unemployment
Adam Reply
what are the causes of unemployment?
Mbubi Reply
The main causes of unemployment are listed below. 1. Frictional unemployment 2. Cyclical unemployment 3. Structural unemployment
We can also categorize the causes on a broader sense as: 1. Political and 2. Social cause As unemployeement root causes are embaded in this two.
would opportunity cost exist if there was no scarcity?
yes just because the opportunity cost arose when there is Alternative to choose among the alternatives.
I am thinking that, if our resources were unlimited, then there wouldn't be any need to forgo some wants. Hence the inexistence if opportunity cost
politics has done what?
consider time assani
I'm Emmanuel,...I taught the main cause is the change in gov't.
...Lack of capital to set up a firm respectively
I would like to bring in Educational levels can also be the cause the cause of the problem respectively
lack of skills among the new generation is the serious issue.
Where I come from , I don't see why education or personal aspects seem to do with unimployment, technically the motivation and eigerness in all works of live is there , dispite the cultural influence and physical bearriors;the thing we lacking is Government Support and open market ethics.
sorry about that-(repation). We have a over powering ethical political system that's displacing the marketing asspects of economy and causing large scale unemployment right across the board...
can someone Explain Expansionary Monetary Policy and Contractionary Monetary Policy Using one of the instrument of Monetary Policy? Please am kinda lost here?. ta
Emmanuel Reply
using a graph show the case of substitute and compliment goods
Ade Reply
can anyone give me a simple explanation to Five Sector Macroeconomics?
Can someone please define what economics is
jason Reply
economics simply is a social science subject that study human behavior.
economics is a social science which studies human behaviour as a relationship between ends and scarce means that has alternative uses
Can someone please tell me how to calculate GDP
emmanual kapal to calculate GDP (Gross Domestic Product) has three method in calculating it (1)income approach (2) expenditure approach (3) value added method
thanks Alae
u are welcome
in basic terms economics is revered to as battery system, it date back to when Men sees the need to exchange sapless goods and produce to gain , either wealth , basic necessities or to establish trading ties for personal benefit or social asspects in terms of coexistence and continuity, future .
what is the law of demand
Berlinda Reply
keep other thing constant, when the price increases demand decrease when the price decreases demand increases of the commodity.
all things being equal,quantity demanded decrease as price increase and increase as price decrease
there's practial joke to it ..." the higher the demand ; scarcity, increase in production and drop in quality"... quite the controversy - for example China vs Europe, United States and we are all boxed up in between somewhere...
Other thing remain constant the low price of commodity the high quantity of commodity and vice versa is true
Explain Effective demand
Anita Reply
What is effective demand
like Modi is in demand...best example of effective demand
Don't get you
Anita you mean you don't get me or who?
level of demand that represents a real intention to purchase by people with the means to pay
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Siyavula textbooks: wiskunde (graad 10) [caps]. OpenStax CNX. Aug 04, 2011 Download for free at http://cnx.org/content/col11328/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Siyavula textbooks: wiskunde (graad 10) [caps]' conversation and receive update notifications?