Exercise:
Von einem Schiff S_ aus sieht man die Spitze eines Leuchtturms unter dem Höhenwinkel alphaang und von einem Schiff S_ aus unter dem Höhenwinkel betaang. Beide Schiffe befinden sich genau westlich vom Leuchtturm und sind m voneinander entfernt. Berechne die Höhe des Leuchtturms. center tikzpicturescale. %Koordinaten coordinate A at ; coordinate B at ; coordinate C at ..; coordinate D at ..; coordinate E at ..; coordinate F at ..; coordinate G at ..; coordinate H at ..; coordinate I at .; coordinate J at .; coordinate K at .; coordinate L at .; coordinate M at .; coordinate N at .; coordinatelabelabove:L O at .; coordinatelabelabove:S_ P at -; coordinatelabelabove:S_ Q at -; %Viereck - Füllung filldrawcolorred! fillred! C -- D -- E -- F -- cycle; filldrawcolorred! fillred! G -- H -- I -- J -- cycle; filldrawcolorblack! fillwhite! K -- L -- N -- M -- cycle; filldrawcolorred! fillred! M -- N -- O -- cycle; filldrawcolorblack! fillwhite! A -- B -- C -- D -- cycle; filldrawcolorblack! fillwhite! E -- F -- G -- H -- cycle; draw colorblack!O -- P; draw colorblack!O -- Q; draw latex-latex colorblack!-- -- -- nodemidway below m; pathangle eccentricity. C -- A -- D pic fill opacity. angle radiuscm right colorred "alpha" draw - fillred! angleA--P--O; pathangle eccentricity. C -- A -- D pic fill opacity. angle radiuscm right colorblue "beta" draw - fillblue! angleA--Q--O; fillNavyBlue plotsmoothdomain-:- samples x.+.*rnd |- --.; fillNavyBlue! plotsmoothdomain-:- samples x+.*rnd |- --.; fillNavyBlue! plotsmoothdomain-:- samples x-.*rnd |- --.; fillNavyBlue! plotsmoothdomain-:- samples x-.*rnd |- --.; fillTan plotsmoothdomain-: samples x.+.*rnd |- --.; draw latex-latex colorblack!. -- . nodemidway right h; tikzpicture center

Solution:
Man berechnet zunächst den Winkel gamma und damit anschliess delta: center tikzpicturescale. %Koordinaten coordinate A at ; coordinate B at ; coordinate C at ..; coordinate D at ..; coordinate E at ..; coordinate F at ..; coordinate G at ..; coordinate H at ..; coordinate I at .; coordinate J at .; coordinate K at .; coordinate L at .; coordinate M at .; coordinate N at .; coordinatelabelabove:L O at .; coordinate P at -; coordinate Q at -; coordinatelabelleft:S_ R at --; coordinatelabelright:S_ S at --; coordinate T at -; %Viereck - Füllung filldrawcolorred! fillred! C -- D -- E -- F -- cycle; filldrawcolorred! fillred! G -- H -- I -- J -- cycle; filldrawcolorblack! fillwhite! K -- L -- N -- M -- cycle; filldrawcolorred! fillred! M -- N -- O -- cycle; filldrawcolorblack! fillwhite! A -- B -- C -- D -- cycle; filldrawcolorblack! fillwhite! E -- F -- G -- H -- cycle; draw colorblack!O -- P; draw colorblack!O -- Q; draw latex-latex colorblack!R -- S nodemidway below m; pathangle eccentricity C -- A -- D pic fill opacity. angle radiuscm right colorred "alpha" draw - fillred! angleA--P--O; pathangle eccentricity C -- A -- D pic fill opacity. angle radius.cm right colorblue "beta" draw - fillblue! angleA--Q--O; pathangle eccentricity. P -- Q -- D pic fill opacity. angle radius.cm colorviolet "gamma" draw - fillviolet! angleO--Q--P; pathangle eccentricity. P -- Q -- D pic fill opacity. angle radiuscm colorviolet below "delta" draw - fillviolet! angleP--O--Q; fillNavyBlue plotsmoothdomain-:- samples x.+.*rnd |- --.; fillNavyBlue! plotsmoothdomain-:- samples x+.*rnd |- --.; fillNavyBlue! plotsmoothdomain-:- samples x-.*rnd |- --.; fillNavyBlue! plotsmoothdomain-:- samples x-.*rnd |- --.; fillTan plotsmoothdomain-: samples x.+.*rnd |- --.; draw latex-latex colorblack!. -- . nodemidway right h; tikzpicture center gamma ang-beta ang delta ang-alpha-gamma ang Nun kann man den Sinussatz verwen um die Strecke S_L zu bestimmen: fracmsindelta fracS_Lsinalpha &Rightarrow S_Lfracmsindelta sinalpha .m Mit diesen Informationen berechnet man schliesslich wie gewünscht h: sinbeta frachS_L &Rightarrow h sinbeta S_L .m Also ist der Leuchtturm .m hoch.