MATHHX A

MATHHX A

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6.2 Cosinus, sinus og tangens

  • Definition 6.2.1
    Lad \(v\) være en vinkel med retningspunkt \(R_v\).

    Cosinus til \(v\) defineres som førstekoordinaten til \(R_v\) og betegnes \(\cos (v)\).

    Sinus til \(v\) defineres som andenkoordinaten til \(R_v\) og betegnes \(\sin (v)\).

    Tangens til \(v\) defineres som forholdet \(\frac {\sin (v)}{\cos (v)}\) og betegnes \(\tan (v)\). Dette gælder selvfølgelig kun hvis \(\cos (v)\neq 0\). Ellers er tangens udefineret.

Cosinus og sinus kan illustreres således:

(-tikz- diagram)

  • Eksempel 6.2.1
    Vi vil bestemme cosinus, sinus og tangens til vinklen på tegningen oven over. Det er svært at se de præcise koordinater til retningspunktet ud fra tegningen, men det ser ud til at cosinus og sinus har samme værdi og at den værdi ligger mellem \(\frac {1}{2}\) og \(1\). Skal vi sige \(0{,}7\)? Altså

    \[\cos (v)=0{,}7\quad \textrm {og}\quad \sin (v)=0{,}7\]

    Tangens bliver så

    \[\tan (v)=\frac {\sin (v)}{\cos (v)}=\frac {0{,}7}{0{,}7}=1\]

Der findes selvfølgelig metoder til at få præcise værdier for cosinus, sinus og tangens, men til at starte med, er det fint at aflæse for at få forståelse på plads

Øvelse 6.2.1

Bestem cosinus, sinus og tangens til følgende vinkler.

  • a) \(v=\pi \)

  • b) \(v=-2\pi \)

  • c) \(v=135\degree \) (bare sådan ca.)

  • d) \(v=-2\) (bare sådan ca.)

  • e) \(v=90\degree \)

Løsning 6.2.1

  • a) \(\cos (\pi )=-1\), \(\sin (\pi )=0\) og \(\tan (\pi )=0\)

  • b) \(\cos (-2\pi )=1\), \(\sin (-2\pi )=0\) og \(\tan (-2\pi )=0\)

  • c) \(\cos (135\degree )=-0{,}7\), \(\sin (135\degree )=0{,}7\) og \(\tan (135\degree )=-1\)

  • d) \(\cos (-2)=-0{,}4\), \(\sin (-2)=-0{,}9\) og \(\tan (-2)=2{,}2\)

  • e) \(\cos (90\degree )=0\), \(\sin (90\degree )=1\) og tangens er ikke defineret.

Cosinus, sinus og tangens som funktioner

Til enhver vinkel \(v\) kan vi bestemme cosinus til \(v\). Vi kan derfor tænke på cosinus som en funktion, nemlig den funktion som til ethvert \(x\) knytter \(\cos (x)\), dvs. funktionen med forskriften \(f(x)=\cos (x)\). Grunden til vi skriver \(x\) i stedet for \(v\) er at vi gerne vil tænke på cosinus som en almindelig funktion og her plejer vi at kalde variablen for \(x\). Lad os undersøge hvordan grafen ser ud. Vi tegner grafen ved at lave et sildeben. Vi finder funktionsværdierne ud fra enhedscirklen. Vi starter med \(x=0\). Altså en vinkel på \(0\).

(-tikz- diagram)

Vi kan se at retningspunktet for vinklen \(x=0\) har en førstekoordinat på \(1\). Altså er \(\cos (0)=1\) og dermed er \(f(0)=1\) (husk at vi tegner grafen for \(f(x)=cos(x)\)). Der er ikke noget nyt her. Det er præcis som den øvelse I lige har regnet. Det skriver vi ind i et sildeben:

\(\begin {array}{|c|c|} \hline x & 0 \\ \hline f(x) & 1 \\ \hline \end {array}\)

Vi tager lige en til. Vi vælger vinklen på \(x=\frac {\pi }{4}\):

(-tikz- diagram)

Vi aflæser førstekoordinaten til retningspunkt for vinklen \(x=\frac {\pi }{4}\) til at være \(x=0{,}7\), hvilket vi sætter ind i sildebenet. Sådan forsætter vi rundt i enhedscirklen, hvilket giver os:

\(\begin {array}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline x & 0 & \frac {\pi }{4} & \frac {\pi }{2} & \frac {3\pi }{4} & \pi & \frac {5\pi }{4} & \frac {3\pi }{2} & \frac {7\pi }{4} & 2\pi \\ \hline f(x) & 1 & 0{,}7 & 0 & - 0{,}7 & -1 & -0{,}7 & 0 & 0{,}7 & 1 \\ \hline \end {array}\)

Vi tegner punkterne ind i et koordinatsystem

(-tikz- diagram)

og vi tegner en graf gennem alle punkterne:

(-tikz- diagram)

Vi ser at det sidste punkt ligger med samme funktionsværdi (\(y\)-værdi ) som det første. Det er fordi at vi ved \(x=2\pi \) er nået en helt gang rundt i cirklen og defor starter vi forfra med de samme funktionsværdier. Tegner vi endnu et punkt ind, ved \(x=2\pi +\frac {\pi }{4}=\frac {9\pi }{4}\), får vi:

(-tikz- diagram)

Vi kan se at dette punkt har samme funktionsværdi som det andet punkt. Sådan kan vi blive ved:

(-tikz- diagram)

Vi kan se at den røde del af grafen er en nøjagtig kopi af den sorte del. En funktion, som gentager sig selv på den måde, siges at være periodisk. Det stykke, man skal gå ud, før at funktion gentager sig selv, kaldes perioden. Vi kan se at cosinus er periodisk med periode \(T=2\pi \). Grafen kan selvfølgelig udvides mere – også i negativ retning:

(-tikz- diagram)

Øvelse 6.2.2

Nu er det din tur til at tegne grafer med papir og blyant som beskrevet ovenover.

  • a) Tegn grafen for sinus i intervallet \([0;2\pi ]\).

  • b) Tegn grafen tangens i intervallet \([0;2\pi ]\). Den her er lidt svær. Du får brug for mange støttepunkter. Det kan være en hjælpe at overveje, hvordan grafen må opfører sig, når \(x\) er tæt på \(\frac {\pi }{2}\) eller \(\frac {3\pi }{2}\).

Løsning 6.2.2

  • a)

    (-tikz- diagram)

    \(f(x)=\sin (x)\)
  • b)

    (-tikz- diagram)

    \(f(x)=\tan (x)\)

Øvelse 6.2.3

  • a) Argumenter for at sinus er periodisk og kom med et bud for perioden. Få hjælp fra din tegning af grafen.

  • b) Argumenter for at tangens er periodisk og kom med et bud for perioden. Få hjælp fra din tegning af grafen.

Løsning 6.2.3

  • a) Perioden for sinus er \(2\pi \)

  • b) Perioden for tangens er \(\pi \).