Sinussatz im Trapez
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Exercise:
Gegeben ist ein Trapez mit den Diagonalen e und f sowie alpha_ang und alpha_ang. center tikzpicturescale. %Koordinaten coordinatelabelbelow left:A A at ; coordinatelabelbelow right:B B at ; coordinatelabelabove right:C C at ; coordinatelabelabove left:D D at ; %Viereck - Füllung fillcolorblack! fillblack! thick A -- B -- C -- D -- cycle; %Viereck - Seiten drawthick A -- B nodemidway below a; drawthick B -- C nodemidway above right b; drawthick C -- D nodemidway above c; drawthick D -- A nodemidway above left d; %Diagonale drawdensely dashed thick colorpurple A -- C nodemidway below slopede; drawdensely dashed thick colorblue!!white B -- D nodemidway below slopedf; %Winkel path C -- A -- D pic fill opacity above left colorpurple "alpha_" draw - thick fillpurple! angleC--A--D; path C -- A -- B pic fill opacity below right colorred "alpha_" draw - thick fillred! angleB--A--C; drawthick A -- B; drawthick A -- D; tikzpicture center Berechne die Seitenlänge a.
Solution:
Man kann zunächst den Gesamtwinkel alpha bestimmen: alpha alpha_+alpha_ ang+ang ang center tikzpicturescale %Koordinaten coordinatelabelbelow left:A A at ; coordinatelabelbelow right:B B at ; coordinatelabelabove right:C C at ; coordinatelabelabove left:D D at ; coordinate E at ; %Viereck - Füllung fillcolorblack! fillblack! thick A -- B -- C -- D -- cycle; %Viereck - Seiten drawthick A -- B nodemidway below a; drawthick B -- C nodemidway above right b; drawthick C -- D nodemidway above c; drawthick D -- A nodemidway above left d; %Diagonale drawdensely dashed thick colorpurple A -- C nodemidway below slopede; drawdensely dashed thick colorblue!!white B -- D nodemidway below slopedf; %Winkel path D -- A -- B pic fill opacity colorred "alpha" draw - thick fillred! angleB--A--D; path A -- D -- E pic fill opacity colorred "alpha" draw - thick fillred! angleE--D--A; path C -- D -- A pic fill opacity colorgreen "beta" draw - thick fillgreen! angleA--D--C; drawthick A -- B; drawthick A -- D; drawthick C -- D; drawthick dashed D -- E; tikzpicture center Damit kann man nun den Winkel beta berechnen: beta ang-alpha ang-alpha_-alpha_ ang-ang-ang ang center tikzpicturescale %Koordinaten coordinatelabelbelow left:A A at ; coordinatelabelbelow right:B B at ; coordinatelabelabove right:C C at ; coordinatelabelabove left:D D at ; coordinate E at ; %Viereck - Füllung fillcolorblack! fillblack! thick A -- B -- C -- D -- cycle; %Viereck - Seiten drawthick A -- B nodemidway below a; drawthick B -- C nodemidway above right b; drawthick C -- D nodemidway above c; drawthick D -- A nodemidway above left d; %Diagonale drawdensely dashed thick colorpurple A -- C nodemidway below slopede; drawdensely dashed thick colorblue!!white B -- D nodemidway below slopedf; %Winkel path D -- A -- B pic fill opacity colorred "alpha" draw - thick fillred! angleB--A--D; path A -- D -- E pic fill opacity colorred "alpha" draw - thick fillred! angleE--D--A; path C -- D -- A pic fill opacity colorgreen "beta" draw - thick fillgreen! angleA--D--C; path D -- C -- A pic fill opacity colorviolet "gamma" draw - thick fillviolet! angleD--C--A; drawthick A -- B; drawthick A -- D; drawthick C -- D; drawthick dashed D -- E; tikzpicture center Man berechnet nun gamma: gamma ang-beta-alpha_ ang-ang-ang ang Anschliess kann man den Sinussatz verwen um d zu ermitteln: center tikzpicturescale %Koordinaten coordinatelabelbelow left:A A at ; coordinatelabelbelow right:B B at ; coordinatelabelabove right:C C at ; coordinatelabelabove left:D D at ; coordinate E at ; %Viereck - Füllung fillcolorblack! fillblack! thick A -- B -- C -- D -- cycle; %Viereck - Seiten drawthick A -- B nodemidway below a; drawthick B -- C nodemidway above right b; drawthick C -- D nodemidway above c; drawthick D -- A nodemidway above left colorviolet d; %Diagonale drawdensely dashed thick colorgreen A -- C nodemidway below slopede; %Winkel path D -- C -- A pic fill opacity colorviolet "gamma" draw - thick fillviolet! angleD--C--A; path C -- D -- A pic fill opacity colorgreen "beta" draw - thick fillgreen! angleA--D--C; drawthick A -- B; drawthick colorviolet A -- D; drawthick C -- D; drawthick dashed D -- E; tikzpicture center fracesinbeta fracdsingamma &Rightarrow dfracesinbeta singamma . Damit kann man nun wieder mit Sinussatz delta bestimmen: center tikzpicturescale %Koordinaten coordinatelabelbelow left:A A at ; coordinatelabelbelow right:B B at ; coordinatelabelabove right:C C at ; coordinatelabelabove left:D D at ; coordinate E at ; %Viereck - Füllung fillcolorblack! fillblack! thick A -- B -- C -- D -- cycle; %Viereck - Seiten drawthick A -- B nodemidway below a; drawthick B -- C nodemidway above right b; drawthick C -- D nodemidway above c; drawthick D -- A nodemidway above left d; %Diagonale drawdensely dashed thick colorpurple A -- C nodemidway below slopede; drawdensely dashed thick colorblue!!white B -- D nodemidway below slopedf; %Winkel path D -- A -- B pic fill opacity colorred "alpha" draw - thick fillred! angleB--A--D; path A -- D -- E pic fill opacity colorred "alpha" draw - thick fillred! angleE--D--A; path C -- D -- A pic fill opacity colorgreen "beta" draw - thick fillgreen! angleA--D--C; path A -- B -- D pic fill opacity below colorblue "delta" draw - thick fillblue! angleD--B--A; drawthick A -- B; drawthick A -- D; drawthick C -- D; drawthick dashed D -- E; tikzpicture center fracdsindelta fracfsinalpha &Rightarrow deltaarccosleftfracd sinalphafright ang. Anschliess bestimmt man den Winkel epsilon mithilfe der Winkelme eines Dreiecks: center tikzpicturescale %Koordinaten coordinatelabelbelow left:A A at ; coordinatelabelbelow right:B B at ; coordinatelabelabove right:C C at ; coordinatelabelabove left:D D at ; coordinate E at ; %Viereck - Füllung fillcolorblack! fillblack! thick A -- B -- C -- D -- cycle; %Viereck - Seiten drawthick A -- B nodemidway below a; drawthick B -- C nodemidway above right b; drawthick C -- D nodemidway above c; drawthick D -- A nodemidway above left d; %Diagonale drawdensely dashed thick colorpurple A -- C nodemidway below slopede; drawdensely dashed thick colorblue!!white B -- D nodemidway below slopedf; %Winkel path D -- A -- B pic fill opacity colorred "alpha" draw - thick fillred! angleB--A--D; path A -- D -- E pic fill opacity colorred "alpha" draw - thick fillred! angleE--D--A; path A -- D -- B pic fill opacity colorTurquoise "epsilon" draw - thick fillTurquoise! angleA--D--B; path A -- B -- D pic fill opacity below colorblue "delta" draw - thick fillblue! angleD--B--A; drawthick A -- B; drawthick A -- D; drawthick C -- D; drawthick dashed D -- E; tikzpicture center epsilon ang-alpha-delta ang. Mit diesen Angaben ist es schliesslich möglich wieder mit Sinussatz die Seitenlänge a zu bestimmen wie in der Aufgabe gefordert: center tikzpicturescale %Koordinaten coordinatelabelbelow left:A A at ; coordinatelabelbelow right:B B at ; coordinatelabelabove right:C C at ; coordinatelabelabove left:D D at ; coordinate E at ; %Viereck - Füllung fillcolorblack! fillblack! thick A -- B -- C -- D -- cycle; %Viereck - Seiten drawthick A -- B nodemidway below colorTurquoise a; drawthick B -- C nodemidway above right b; drawthick C -- D nodemidway above c; drawthick D -- A nodemidway above left d; %Diagonale drawdensely dashed thick colorblue!!white B -- D nodemidway below slopedf; %Winkel path D -- A -- B pic fill opacity colorblue "alpha" draw - thick fillblue! angleB--A--D; path A -- D -- B pic fill opacity colorTurquoise "epsilon" draw - thick fillTurquoise! angleA--D--B; drawthick colorTurquoise A -- B; drawthick A -- D; drawthick C -- D; tikzpicture center fracfsinalpha fracasinepsilon &Rightarrow afracfsinalpha sinepsilon .
Gegeben ist ein Trapez mit den Diagonalen e und f sowie alpha_ang und alpha_ang. center tikzpicturescale. %Koordinaten coordinatelabelbelow left:A A at ; coordinatelabelbelow right:B B at ; coordinatelabelabove right:C C at ; coordinatelabelabove left:D D at ; %Viereck - Füllung fillcolorblack! fillblack! thick A -- B -- C -- D -- cycle; %Viereck - Seiten drawthick A -- B nodemidway below a; drawthick B -- C nodemidway above right b; drawthick C -- D nodemidway above c; drawthick D -- A nodemidway above left d; %Diagonale drawdensely dashed thick colorpurple A -- C nodemidway below slopede; drawdensely dashed thick colorblue!!white B -- D nodemidway below slopedf; %Winkel path C -- A -- D pic fill opacity above left colorpurple "alpha_" draw - thick fillpurple! angleC--A--D; path C -- A -- B pic fill opacity below right colorred "alpha_" draw - thick fillred! angleB--A--C; drawthick A -- B; drawthick A -- D; tikzpicture center Berechne die Seitenlänge a.
Solution:
Man kann zunächst den Gesamtwinkel alpha bestimmen: alpha alpha_+alpha_ ang+ang ang center tikzpicturescale %Koordinaten coordinatelabelbelow left:A A at ; coordinatelabelbelow right:B B at ; coordinatelabelabove right:C C at ; coordinatelabelabove left:D D at ; coordinate E at ; %Viereck - Füllung fillcolorblack! fillblack! thick A -- B -- C -- D -- cycle; %Viereck - Seiten drawthick A -- B nodemidway below a; drawthick B -- C nodemidway above right b; drawthick C -- D nodemidway above c; drawthick D -- A nodemidway above left d; %Diagonale drawdensely dashed thick colorpurple A -- C nodemidway below slopede; drawdensely dashed thick colorblue!!white B -- D nodemidway below slopedf; %Winkel path D -- A -- B pic fill opacity colorred "alpha" draw - thick fillred! angleB--A--D; path A -- D -- E pic fill opacity colorred "alpha" draw - thick fillred! angleE--D--A; path C -- D -- A pic fill opacity colorgreen "beta" draw - thick fillgreen! angleA--D--C; drawthick A -- B; drawthick A -- D; drawthick C -- D; drawthick dashed D -- E; tikzpicture center Damit kann man nun den Winkel beta berechnen: beta ang-alpha ang-alpha_-alpha_ ang-ang-ang ang center tikzpicturescale %Koordinaten coordinatelabelbelow left:A A at ; coordinatelabelbelow right:B B at ; coordinatelabelabove right:C C at ; coordinatelabelabove left:D D at ; coordinate E at ; %Viereck - Füllung fillcolorblack! fillblack! thick A -- B -- C -- D -- cycle; %Viereck - Seiten drawthick A -- B nodemidway below a; drawthick B -- C nodemidway above right b; drawthick C -- D nodemidway above c; drawthick D -- A nodemidway above left d; %Diagonale drawdensely dashed thick colorpurple A -- C nodemidway below slopede; drawdensely dashed thick colorblue!!white B -- D nodemidway below slopedf; %Winkel path D -- A -- B pic fill opacity colorred "alpha" draw - thick fillred! angleB--A--D; path A -- D -- E pic fill opacity colorred "alpha" draw - thick fillred! angleE--D--A; path C -- D -- A pic fill opacity colorgreen "beta" draw - thick fillgreen! angleA--D--C; path D -- C -- A pic fill opacity colorviolet "gamma" draw - thick fillviolet! angleD--C--A; drawthick A -- B; drawthick A -- D; drawthick C -- D; drawthick dashed D -- E; tikzpicture center Man berechnet nun gamma: gamma ang-beta-alpha_ ang-ang-ang ang Anschliess kann man den Sinussatz verwen um d zu ermitteln: center tikzpicturescale %Koordinaten coordinatelabelbelow left:A A at ; coordinatelabelbelow right:B B at ; coordinatelabelabove right:C C at ; coordinatelabelabove left:D D at ; coordinate E at ; %Viereck - Füllung fillcolorblack! fillblack! thick A -- B -- C -- D -- cycle; %Viereck - Seiten drawthick A -- B nodemidway below a; drawthick B -- C nodemidway above right b; drawthick C -- D nodemidway above c; drawthick D -- A nodemidway above left colorviolet d; %Diagonale drawdensely dashed thick colorgreen A -- C nodemidway below slopede; %Winkel path D -- C -- A pic fill opacity colorviolet "gamma" draw - thick fillviolet! angleD--C--A; path C -- D -- A pic fill opacity colorgreen "beta" draw - thick fillgreen! angleA--D--C; drawthick A -- B; drawthick colorviolet A -- D; drawthick C -- D; drawthick dashed D -- E; tikzpicture center fracesinbeta fracdsingamma &Rightarrow dfracesinbeta singamma . Damit kann man nun wieder mit Sinussatz delta bestimmen: center tikzpicturescale %Koordinaten coordinatelabelbelow left:A A at ; coordinatelabelbelow right:B B at ; coordinatelabelabove right:C C at ; coordinatelabelabove left:D D at ; coordinate E at ; %Viereck - Füllung fillcolorblack! fillblack! thick A -- B -- C -- D -- cycle; %Viereck - Seiten drawthick A -- B nodemidway below a; drawthick B -- C nodemidway above right b; drawthick C -- D nodemidway above c; drawthick D -- A nodemidway above left d; %Diagonale drawdensely dashed thick colorpurple A -- C nodemidway below slopede; drawdensely dashed thick colorblue!!white B -- D nodemidway below slopedf; %Winkel path D -- A -- B pic fill opacity colorred "alpha" draw - thick fillred! angleB--A--D; path A -- D -- E pic fill opacity colorred "alpha" draw - thick fillred! angleE--D--A; path C -- D -- A pic fill opacity colorgreen "beta" draw - thick fillgreen! angleA--D--C; path A -- B -- D pic fill opacity below colorblue "delta" draw - thick fillblue! angleD--B--A; drawthick A -- B; drawthick A -- D; drawthick C -- D; drawthick dashed D -- E; tikzpicture center fracdsindelta fracfsinalpha &Rightarrow deltaarccosleftfracd sinalphafright ang. Anschliess bestimmt man den Winkel epsilon mithilfe der Winkelme eines Dreiecks: center tikzpicturescale %Koordinaten coordinatelabelbelow left:A A at ; coordinatelabelbelow right:B B at ; coordinatelabelabove right:C C at ; coordinatelabelabove left:D D at ; coordinate E at ; %Viereck - Füllung fillcolorblack! fillblack! thick A -- B -- C -- D -- cycle; %Viereck - Seiten drawthick A -- B nodemidway below a; drawthick B -- C nodemidway above right b; drawthick C -- D nodemidway above c; drawthick D -- A nodemidway above left d; %Diagonale drawdensely dashed thick colorpurple A -- C nodemidway below slopede; drawdensely dashed thick colorblue!!white B -- D nodemidway below slopedf; %Winkel path D -- A -- B pic fill opacity colorred "alpha" draw - thick fillred! angleB--A--D; path A -- D -- E pic fill opacity colorred "alpha" draw - thick fillred! angleE--D--A; path A -- D -- B pic fill opacity colorTurquoise "epsilon" draw - thick fillTurquoise! angleA--D--B; path A -- B -- D pic fill opacity below colorblue "delta" draw - thick fillblue! angleD--B--A; drawthick A -- B; drawthick A -- D; drawthick C -- D; drawthick dashed D -- E; tikzpicture center epsilon ang-alpha-delta ang. Mit diesen Angaben ist es schliesslich möglich wieder mit Sinussatz die Seitenlänge a zu bestimmen wie in der Aufgabe gefordert: center tikzpicturescale %Koordinaten coordinatelabelbelow left:A A at ; coordinatelabelbelow right:B B at ; coordinatelabelabove right:C C at ; coordinatelabelabove left:D D at ; coordinate E at ; %Viereck - Füllung fillcolorblack! fillblack! thick A -- B -- C -- D -- cycle; %Viereck - Seiten drawthick A -- B nodemidway below colorTurquoise a; drawthick B -- C nodemidway above right b; drawthick C -- D nodemidway above c; drawthick D -- A nodemidway above left d; %Diagonale drawdensely dashed thick colorblue!!white B -- D nodemidway below slopedf; %Winkel path D -- A -- B pic fill opacity colorblue "alpha" draw - thick fillblue! angleB--A--D; path A -- D -- B pic fill opacity colorTurquoise "epsilon" draw - thick fillTurquoise! angleA--D--B; drawthick colorTurquoise A -- B; drawthick A -- D; drawthick C -- D; tikzpicture center fracfsinalpha fracasinepsilon &Rightarrow afracfsinalpha sinepsilon .