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Trigonometri – retvinklet trekant (1:3). Beregning af sidelængder ved Pythagoras.

Formålet med denne artikelserie er, at gøre den studerende i stand til at løse eksamensopgaver omhandlende retvinklede trekanter i forbindelse med den skriftlige matematikeksamen på niveau c. Vi vil i denne artikel give dig løsningen til, hvordan opgavespørgsmål i retvinklede trekanter skal gribes an.

Vi gennemgår i artikelserien de typeopgaver, du kan blive stillet overfor til den skriftlige matematikeksamen. Du vil i hovedtræk kunne komme ud for følgende, når du skal løse en eksamensopgave omkring retvinklede trekanter:

  • Beregning af sidelængder i den retvinklede trekant ved hjælp af Pythagoras
  • Beregning af vinkler og sidelængder i den retvinklede trekant ved hjælp af sinus, cosinus og tangens.
  • Beregning af den retvinklede trekants areal

Hvis man holder ovenstående for øje, vil man på relativt hurtigt kunne lære fremgangsmåden til hvordan typeopgaverne til eksamen skal gribes an. Vi vil i denne artikel gennemgå hvordan man beregner sidelængder ved hjælp af Pythagoras´ læresætning. Vi anvender eksempler som gerne skal bidrage til forståelsen.

Eksempel på løsning af eksamensopgave i retvinklede trekanter

Herunder er afbildet en retvinklet trekant ABC, hvor nogle af trekantens mål er oplyst.

Retvinklet trekant 1 artikel

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Beregning af  sidelængder ved hjælp af Pythagoras

I ovenstående retvinklede trekant kan det ses, at længden af BC = 5 og længden af siden AC = 8, og man skal bestemme længden af AB. For at kunne løse spørgsmålet, skal man gøre sig klart, hvad man vil bruge for at finde længden af AB, når man har fået oplyst trekantens to kateter.
For at kunne finde længden af AB kan man anvende Pythagoras´ læresætning, da man kender to sider i den retvinklet trekant. Pythagoras´ læresætning er;

a2 + b2 = c2

hvor a og b er trekantens kateter, og c er trekantens hypotenuse. Hypotenusen er kendetegnet ved, at være trekantens længste side (i vores eksempel er det denne side vi ikke kender). Hvis man ønsker at beregne længden af AB (hypotenusen), så kan man derfor indsætte vores værdier i Pythagoras´ læresætning, hvorefter vi løser ligningen med en ubekendt. Vi starter først med at navnegive trekantens sider. Ud fra vinkel A bliver modstående sidestykke lille a osv. Se figuren herunder;
Retvinklet trekant 2 artikel

Da vi nu har defineret sidestykkerne i trekanten, kan vi indsætte værdierne i Pythagoras´ læresætning og beregne længden af AB. Beregningen ses herunder;

 

a2 + b2 = c2                                               =>

52 + 82 = c2                                               =>

25 + 64 = c2                                              =>

89 = c2                                                      =>
Retvinklet trekant 3 artikel                                         =>
c = 9,43

 

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Når man kender to sidelængder i en retvinklet trekant, så vil man altid kunne beregne den ukendte sidelængde ved hjælp af Pythagoras.

Hvis vi nu ændrer lidt på vores eksempel, så vi nu kun kender sidelængderne BC = 5 (en katete) og AB = 9,43 (hypotenusen), og skal bestemme sidelængden AC (katete), kan vi igen anvende Pythagoras´ læresætning for at finde den ukendte sidelængde.

Retvinklet trekant 4 artikel
Vi starter igen med, at definerer sidestykkerne i trekanten ved at navngive dem, så det modstående sidestykke ud fra vinkel A bliver lille a osv. Det er gjort i figuren herunder;
Retvinklet trekant 5 artikel

 

Vi har nu fået defineret sidestykkerne, hvorefter vi indsætter i Pythagoras for at finde længden af AC;

a2 + b2 = c2                                             =>

52 + b2 = 9,432                                        =>

25 + b2 = 88,925                                     =>

b2 = 88,925 – 25                                      =>
Retvinklet trekant 6 artikel                   =>
b = 8

Igen ændrer vi lidt på vores eksempel, så vi nu kun kender sidelængderne AC = 8 (en katete) og sidelængden AB = 9,43 (hypotenusen), og skal beregne sidelængden BC (katete). Vi anvender igen Pythagoras for at finde den ukendte sidelængde.

Retvinklet trekant 7 artikel

For at finde sidelængden BC definerer vi igen sidestykkerne i trekanten, så modstående sidestykke ud fra vinkel B bliver lille b osv. Det er vist i figuren herunder;
Retvinklet trekant 8 artikel

Sidestykkerne er defineret, og vi indsætter i Pythagoras for at bestemme sidelængden BC;

a2 + b2 = c2                                             =>

a2 + 82 = 9,432                                        =>

a2 + 64 = 88,925                                     =>

a2 = 88,925 – 64                                      =>
Retvinklet trekant 9 artikel                   =>
b = 5

 

Vil du vide mere om Pythagoras?

Her har vi sørget for du kan læse mere om Pythagoras og hans historie. Her kan du samtidigt finde flere links til andre relevante og spændende artikler.

Herunder har vi samlet endnu flere artikler, som kan være relevante for dig:
Sådan får du topkarakter i matematik, Lær at blive den bedste til potens funktion, Få styr på renters rente – kapitalfremskrivning!

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