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I Zestigdelige graad
1 Definitie
Een volledige cirkelomtrek correspondeert met \(360^\circ \).
\[ 1^\circ = 60' \quad \text {en} \quad 1' = 60'' .\]
2 GRM
“DEG” of “DEGREE”.
TI-84: [MODE], kies DEGREE. De symbolen \(^\circ \), \('\) en \(''\) vind je onder
[2nd] [APPS] (ANGLE).
II Radialen
1 Definitie
De hoekgrootte van de middelpuntshoek waarvan de bijhorende boog precies even lang is als de straal van de cirkel.
2 Berekeningen
\[ P_{\text {cirkelomtrek}} = 2\pi r \]
\[ 360^\circ = 2\pi \,\text {rad} \]
\[ \Rightarrow 180^\circ = \pi \,\text {rad} \]
3 GRM
“RAD” of “RADIAN”
TI-84: [MODE], kies RADIAN.
4 Omzettingsformules
\(^\circ \) \('\) \(''\) in rad:
\(\seteqnumber{0}{}{0}\)
\begin{align*}
180^\circ &= \pi \,\text {rad} \\ \Rightarrow \quad 1^\circ &= \frac {\pi }{180}\,\text {rad} \\ \Rightarrow \quad 1^\circ &\approx 0{,}01745\,\text {rad}
\end{align*}
rad wordt decimaal aangeduid!
\[ \boxed {\,p^\circ =\dfrac {\pi \,p}{180}\,\text {rad}\,} \]
rad in \(^\circ \) \('\) \(''\):
\(\seteqnumber{0}{}{0}\)
\begin{align*}
\pi \,\text {rad} &= 180^\circ \\ \Rightarrow \quad 1\,\text {rad} &= \frac {180^\circ }{\pi } \\ \Rightarrow \quad 1\,\text {rad} &\approx 57^\circ \,17'\,45''
\end{align*}
\[ \boxed {\,p\,\text {rad}=\dfrac {p\cdot 180^\circ }{\pi }\,} \]
5 Overzicht merkwaardige hoeken
We duiden ze aan op een cirkel. Bijvoorbeeld
\(\seteqnumber{0}{}{0}\)
\begin{align*}
30^\circ &= \frac {\pi \cdot 30}{180}\,\text {rad} = \frac {\pi }{6}\,\text {rad}.
\end{align*}
6 Afspraak
Als er geen hoekeenheid vermeld staat, dan is de hoek uitgedrukt in radialen.
bv.
\[ \sin 1 \text { rad} = \sin 1\]
Voor zestigdelige graad schrijven we steeds de graden, minuten en seconden. De maximale nauwkeurigheid is 1 seconde.
We zullen hoeken in de mate van het mogelijke herleiden tot het interval \([0^\circ , 360^\circ [\), of in radialen \([0, 2\pi [\). Soms
wordt ook het interval \(]-\pi , \pi ]\) gebruikt.
III Oefeningen
VBTL - Analyse 1b - p131
nr 1, 2, 3, 4 (van elk 2)
nr 5 of 6 (voorbeeld doen)
niet nr 7 (pas na volgend hoofdstuk)
Opgave p131 nr7. De lengte van de boog \(\wideparen {ACB}\) verhoudt zich tot de lengte van de boog \(\wideparen {ADB}\)
zoals \(3\) tot \(5\). Bepaal in radialen de grootte van de hoek \(\widehat {AOB}\), die de boog \(\wideparen {ACB}\) onderspant.
Uitwerking.
Samen vormen beide bogen de volledige cirkelomtrek, die overeenkomt met \(2\pi \,\text {rad}\). De omtrek wordt dus verdeeld in
\(3+5=8\) gelijke delen, waarvan de boog \(\wideparen {ACB}\) er \(3\) inneemt:
\[ \wideparen {ACB} = \frac {3}{8}\cdot 2\pi \,\text {rad} = \frac {3\pi }{4}\,\text {rad}. \]
Een middelpuntshoek is even groot als de boog die hij onderspant, dus
\[ \boxed {\,\widehat {AOB} = \frac {3\pi }{4}\,\text {rad} \approx 2{,}356\,\text {rad}\,} \]
Ter controle: \(\dfrac {3\pi }{4}\,\text {rad} = \dfrac {3\pi }{4}\cdot \dfrac {180^\circ }{\pi } = 135^\circ \), en de
boog \(\wideparen {ADB}\) komt overeen met \(225^\circ \), zodat \(135^\circ : 225^\circ = 3 : 5\).
Opgave p131 nr6. De wielen van de fiets van Wouter hebben een straal van \(30\,\text {cm}\). Als Wouter zijn fiets
\(75\,\text {cm}\) verplaatst, over welke hoek zijn de wielen dan gedraaid? Bereken de grootte van die hoek in radialen en zestigdelige graden.
Uitwerking.
Het wiel rolt zonder glijden: de afgelegde weg \(s\) is precies even lang als de boog die het wiel afrolt. Per definitie van de radiaal geldt
\[ \alpha = \frac {s}{r}\,\text {rad}. \]
Dus
\[ \boxed {\;\alpha = \frac {75\,\text {cm}}{30\,\text {cm}} = 2{,}5\,\text {rad}\;} \]
Omzetten naar zestigdelige graden:
\(\seteqnumber{0}{}{0}\)
\begin{align*}
2{,}5\,\text {rad} &= \frac {2{,}5\cdot 180^\circ }{\pi } \approx 143{,}2394^\circ \\ &\approx 143^\circ + 0{,}2394\cdot 60' \approx 143^\circ \,14{,}37' \\ &\approx
143^\circ \,14'\,22''.
\end{align*}
\[ \boxed {\;\alpha \approx 143^\circ \,14'\,22''\;} \]
Merk op dat dit iets minder is dan \(\frac {75}{2\pi \cdot 30}\approx 0{,}40\) omwenteling.
