MATHHX A
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6.1 Enhedscirklen
En enhedscirkel er en cirkel i et koordinatsystem med centrum i \((0,0)\) og radius \(r=1\):
Vi får brug for at kende omkredsen af enhedscirklen. Den er \(2\pi \).
Vinkler i enhedscirklen
Har man en vinkel \(v\), tegner man den ind i enhedscirklen som vist her:
Øvelse 6.1.3
Find papir og blyant frem.
a) Tegn en enhedscirkel.
b) Tegn vinklen \(v=135\degree \) ind i enhedscirklen, og inden du siger: ”Jeg har jo ikke nogen vinkelmåler med”, så siger jeg: ”Det er ikke nødvendigt,
du kan godt gøre det præcist nok uden”.
Negative vinkler
Vi er vandt til vinkler er positive. Vinkler er jo noget, vi måler med vores vinkelmåler, right? Men man kan faktisk godt have negative vinkler. Har vi en vinkel på \(v=-45\degree \), betyder det at den skal se ud som en vinkel på
\(45\degree \), men at den skal ”rotere den anden vej” i enhedscirklen som vist her:
Man kunne tænke at en vinkel på \(-45\degree \) er det samme som en vinkel på \(315\degree \) som vist her:
Selvom vinklernes ben ligger samme sted i de to enhedscirkler, kan vi se at den sorte buede pil der markerer rotationen er forskellig for de to vinkler. Spørger man mig, ja så vil jeg hellere roteres med \(-45\degree \) end
\(315\degree \), da \(315\degree \) ville gøre mig helt rundtosset (det er et større rotation). De to vinkler har dog en noget til fælles. De skærer enhedscirklen i det samme punkt, og netop skæringspunktet med enhedscirklen
kommer vi til at interessere os for senere.
Retningspunkter
Når man tegner en vinkel ind i enhedscirklen, ligger man altid vinklens ene ben langs x-aksen. Det punkt hvor vinklens andet ben skærer cirklen kaldes retningspunktet . Hedder vinklen \(v\), betegner vi retningspunktet
med \(R_v\).
Øvelse 6.1.5
Bestem koordinaterne til retningspunkterne for følgende vinkler.
a) \(v=90\degree \)
b) \(v=180\degree \)
c) \(v=270\degree \)
d) \(v=0\degree \)
e) \(v=-90\degree \)
Løsning 6.1.5
a) \(R_v=(0,1)\)
b) \(R_v=(-1,0)\)
c) \(R_v=(0,-1)\)
d) \(R_v=(1,0)\)
e) \(R_v=(0,-1)\)
Vinkler målt i radianer
I stedet for at måle vinkler i grader, kan man måle dem i noget som hedder radianer . Når man måler en vinkel i radianer, måler man længden langs cirklens periferi (kant) som vist her:
Vinklen \(v\) målt i radianer er altså længden af den grønne bue. Størrelsen på en vinkel målt i radianer kaldes radiantallet.
Eksempel 6.1.1
Vi vil nu bestemme radiantallet for vinklen ovenover: Da der er \(2\pi \) hele vejen rundt, og vi kan se at den grønne bue fylder \(\frac {1}{8}\) af cirklen, må vinklen \(v\) være
\(v=\frac {1}{8}\cdot 2\pi =\frac {\pi }{4}\). Altså \(v=\frac {\pi }{4}\).
Læg mærke til der ikke er nogen enhed på. Det er den måde vi kan se forskel på grader og radianer. Står der \(v=20\), betyder det \(20\) radianer, mens \(v=20\degree \) betyder 20 grader.
Vil man finde radiantallet for negative vinkler, fungerer det helt som man skulle tro. Man finder stadig buelængden, men hvis vinklen er negativ, så er radiantallet også negativt.
Eksempel 6.1.2
Vi vil nu bestemme radiantallet for en vinkel på \(-90\degree \). Vi indtegner først vinklen og derefter indtegner vi buen langs enhedscirklen (igen markeret med grøn):
Da der er \(2\pi \) hele vejen rundt, og vi kan se at den grønne bue fylder \(\frac {1}{4}\) af cirklen, må buelængden være \(v=\frac {1}{4}\cdot 2\pi =\frac {\pi }{2}\). Da \(v\) er en negativ vinklen skal der et
negativt fortegn på buelængen. Altså \(v=-\frac {\pi }{2}\).
Øvelse 6.1.6
Bestem radiantallet for følgende vinker:
a) \(v=90\degree \)
b) \(v=180\degree \)
c) \(v=0\degree \)
d) \(v=-45\degree \)
Øvelse 6.1.7
Bestem koordinaterne til retningspunktet for følgende vinkler. Pas på! Jeg har blandet grader og radianer.
Løsning 6.1.8
a) Sådan her.
\(\begin {array}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline x & 0\degree & 45\degree & 90\degree & 135 \degree & 180\degree & 225\degree & 270\degree & 315\degree &
360\degree \\ \hline x & 0 & \frac {\pi }{4} & \frac {\pi }{2} & \frac {3\pi }{4} & \pi & \frac {5\pi }{4} & \frac {3\pi }{2} & \frac {7\pi }{4} & 2\pi \\
\hline \end {array}\)