De sinusregel

Antwoorden bij de opgaven

1. ${\displaystyle \frac{4}{sin\left({50}^{\text{o}}\right)}=\frac{\left|AB\right|}{sin\left({110}^{\text{o}}\right)}}$ geeft ${\displaystyle \left|AB\right|\approx \mathrm{4,907}}$.
   ${\displaystyle \frac{4}{sin\left({50}^{\text{o}}\right)}=\frac{\left|BP\right|}{sin\left({20}^{\text{o}}\right)}}$ geeft ${\displaystyle \left|BP\right|\approx \mathrm{1,786}}$.
   Je loopt dus ${\displaystyle 4+\mathrm{1,786}-\mathrm{4,907}=\mathrm{0,879}}$ km om.
2. ${\displaystyle \frac{4}{sin\left({70}^{\text{o}}\right)}=\frac{\left|BC\right|}{sin\left({30}^{\text{o}}\right)}}$ geeft ${\displaystyle \left|BC\right|\approx \mathrm{2,13}}$.
3. ${\displaystyle \frac{4}{sin\left({80}^{\text{o}}\right)}=\frac{\left|DF\right|}{sin\left({70}^{\text{o}}\right)}}$ geeft ${\displaystyle \left|DF\right|\approx \mathrm{3,82}}$.
4. Trek de hoogtelijn uit ${\displaystyle B}$ op ${\displaystyle AC}$. Die hoogtelijn kun je op twee manieren berekenen: ${\displaystyle a\cdot sin\left(\gamma \right)=c\cdot sin\left(\alpha \right)}$.
5. 1. Doen.
   2. Omdat je bij het toepassen van de sinusregel altijd één van de drie breuken compleet moet weten en in elke breuk een zijde en de tegenoverliggende hoek voorkomen.
   3. Doen.
6. ${\displaystyle \left|AB\right|\approx \mathrm{34,1}}$; ${\displaystyle \left|EF\right|\approx \mathrm{76,9}}$; ${\displaystyle \left|HI\right|\approx \mathrm{60,7}}$; ${\displaystyle \left|MN\right|\approx \mathrm{27,1}}$
7. Bijvoorbeeld ${\displaystyle a\cdot sin\left(\beta \right)=b\cdot sin\left(\alpha \right)}$ of ${\displaystyle a=b\cdot \frac{sin\left(\alpha \right)}{sin\left(\beta \right)}}$.
8. 1. ${\displaystyle \frac{6}{sin\left({50}^{\text{o}}\right)}=\frac{\left|AC\right|}{sin\left({90}^{\text{o}}\right)}}$ geeft ${\displaystyle \left|AC\right|\approx \mathrm{7,8}}$.
   2. ${\displaystyle cos\left({40}^{\text{o}}\right)=\frac{6}{\left|AC\right|}}$geeft ${\displaystyle \left|AC\right|\approx \mathrm{7,8}}$.
9. 1. Gebruik de sinusregel.
   2. ${\displaystyle \angle C\approx {\mathrm{41,8}}^{\text{o}}}$ en ${\displaystyle \angle C\approx {\mathrm{138,2}}^{\text{o}}}$
   3. Alleen ${\displaystyle \angle C\approx {\mathrm{41,8}}^{\text{o}}}$ kan.
   4. ${\displaystyle \left|AC\right|\approx \mathrm{29,4}}$
10. ${\displaystyle \left|AB\right|\approx \mathrm{5,4}}$; ${\displaystyle \left|DE\right|=\frac{8}{3}\sqrt{6}}$; ${\displaystyle \left|DF\right|\approx \mathrm{8,9}}$; ${\displaystyle \left|HI\right|=4\sqrt{3}}$
11. ${\displaystyle \left|AB\right|\approx \mathrm{306,4}}$ m.
12. 1. Doen
    2. ${\displaystyle \mathrm{23,46}+\mathrm{1,80}\approx 25}$ m
    3. ${\displaystyle \frac{h}{\left|BC\right|+17}=\frac{6}{9}}$ en ${\displaystyle \frac{h}{\left|BC\right|}=\frac{6}{8}}$ geeft ${\displaystyle h\approx 102}$ m.
13. ${\displaystyle \left|AP\right|\approx \mathrm{217,0}}$ m en ${\displaystyle \left|BP\right|\approx \mathrm{206,6}}$ m.
14. Sinusregel in ${\displaystyle \Delta ABC}$ geeft ${\displaystyle \angle C\approx {38}^{\text{o}}}$ en daarmee ${\displaystyle \angle A\approx {74}^{\text{o}}}$. Nogmaals de sinusregel geeft ${\displaystyle \left|BC\right|\approx \mathrm{6,21}}$.
    In ${\displaystyle \Delta DEF}$ kan alles exact: ${\displaystyle \left|DF\right|=4+4\sqrt{3}}$ en ${\displaystyle \left|EF\right|=4\sqrt{6}}$.
15. Bereken eerst met de sinusregel dat ${\displaystyle \angle A\approx {62}^{\text{o}}}$ of ${\displaystyle \angle A\approx {118}^{\text{o}}}$.
    Dan is ${\displaystyle \angle C\approx {74}^{\text{o}}}$ of ${\displaystyle \angle C\approx {27}^{\text{o}}}$.
    En daarmee vind je ${\displaystyle \left|AB\right|\approx \mathrm{5,4}}$ of ${\displaystyle \left|AB\right|\approx \mathrm{2,6}}$.