Trigonometrie Basics
About points...
We associate a certain number of points with each exercise.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.
That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.
That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
About difficulty...
We associate a certain difficulty with each exercise.
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
Question
Solution
Short
Video
\(\LaTeX\)
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Exercise:
Betrachte das folge rechtwinklige Dreieck. center RechtwinkligesDreieckMitWinkel center Bestimme jeweils die gesuchte Grösse. Schreibe immer zuerst allgemein den Winkelsatz stelle ihn algebraisch nach der gesuchten Grösse um setze die gegebenen Grössen ein und berechne dann das Ergebnis! nprvmulticols abclist %%% newcommandgegges tabularrl Gegeben: & ##^circ ## Gesucht: & # tabular %%% abc geggesalphac.a abc geggesbetab.a abc geggesalphab.c abc geggesbetac.a abc geggesalphaac abc geggesbetab.c abc geggesalphac.b abc geggesbetaac abc geggesalphaa.b abc geggesbetaa.b abc geggesalphab.a abc geggesbetacb %%% renewcommandgegges tabularrl Gegeben: & ## ## Gesucht: & # tabular %%% abc gegges acalpha abc gegges b.c.alpha abc gegges a.b.beta abc gegges a.b.alpha abc gegges b.c.beta abc gegges a.c.beta abclist nprvmulticols
Solution:
newcommandANboxedmathttscriptstyle N^- newcommandACOSboxedmathttscriptstyle COS^- newcommandATANboxedmathttscriptstyle TAN^- abclist abc a: Gegenkathete von alpha c: Hypotenuse Rightarrow sinalpha * sinalpha frac ac && | c a c sinalpha . sin^circ .. * abc a: Ankathete von beta b: Gegenkathete von beta Rightarrow tanbeta * tanbeta frac ba && | a :tanbeta a fracbtanbeta frac.tan^circ .. * abc b: Ankathete von alpha c: Hypotenuse Rightarrow cosalpha * cosalpha frac bc && | c : cosalpha c fracbcosalpha frac.cos^circ .. * abc a: Ankathete von beta c: Hypotenuse Rightarrow cosbeta * cosbeta frac ac && | c a c cosbeta . cos^circ .. * abc a: Gegenkathete von alpha c: Hypotenuse Rightarrow sinalpha * sinalpha frac ac && | c:sinalpha c frac a sinalpha frac sin^circ . * abc b: Gegenkathete von beta c: Hypotenuse Rightarrow cosbeta * sinbeta frac bc && | c:sinbeta c fracbsinbeta frac.sin^circ .. * abc b: Ankathete von alpha c: Hypotenuse Rightarrow cosalpha * cosalpha frac bc && | c b c cosalpha . cos^circ .. * abc a: Ankathete von beta c: Hypotenuse Rightarrow cosbeta * cosbeta frac ac && | c :cosbeta c fraca cosbeta frac cos^circ . * abc a: Gegenkathete von alpha b: Ankathete von alpha Rightarrow tanalpha * tanalpha frac ab && | b :tanalpha b fracatanalpha frac.tan^circ .. * abc a: Ankathete von beta b: Gegenkathete von beta Rightarrow tanbeta * tanbeta frac ba && | a b a tanbeta . tan^circ .. * abc a: Gegenkathete von alpha b: Ankathete von alpha Rightarrow tanalpha * tanalpha frac ab && | b a b tanalpha . tan^circ .. * abc b: Gegenkathete von beta c: Hypotenuse Rightarrow sinbeta * sinbeta frac bc && | c b c sinbeta sin^circ . * abc a: Gegenkathete von alpha c: Hypotenuse Rightarrow sinalpha * sinalpha frac ac frac &&|AN alpha ^circ. * abc b: Ankathete von alpha c: Hypotenuse Rightarrow cosalpha * cosalpha frac bc frac.. &&|ACOS alpha .^circ. * abc a: Ankathete von beta b: Gegenkathete von beta Rightarrow tanbeta * tanbeta frac ba frac .. &&|ATAN beta .^circ. * abc a: Gegenkathete von alpha b: Ankathete von alpha Rightarrow tanalpha * tanalpha frac ab frac.. &&|ATAN alpha .^circ. * abc b: Gegenkathete von beta c: Hypotenuse Rightarrow sinbeta * sinbeta frac bc frac.. &&|AN beta .^circ. * abc a: Ankathete von beta c: Hypotenuse Rightarrow cosbeta * cosbeta frac ac frac.. &&|AN beta .^circ. * abclist
Betrachte das folge rechtwinklige Dreieck. center RechtwinkligesDreieckMitWinkel center Bestimme jeweils die gesuchte Grösse. Schreibe immer zuerst allgemein den Winkelsatz stelle ihn algebraisch nach der gesuchten Grösse um setze die gegebenen Grössen ein und berechne dann das Ergebnis! nprvmulticols abclist %%% newcommandgegges tabularrl Gegeben: & ##^circ ## Gesucht: & # tabular %%% abc geggesalphac.a abc geggesbetab.a abc geggesalphab.c abc geggesbetac.a abc geggesalphaac abc geggesbetab.c abc geggesalphac.b abc geggesbetaac abc geggesalphaa.b abc geggesbetaa.b abc geggesalphab.a abc geggesbetacb %%% renewcommandgegges tabularrl Gegeben: & ## ## Gesucht: & # tabular %%% abc gegges acalpha abc gegges b.c.alpha abc gegges a.b.beta abc gegges a.b.alpha abc gegges b.c.beta abc gegges a.c.beta abclist nprvmulticols
Solution:
newcommandANboxedmathttscriptstyle N^- newcommandACOSboxedmathttscriptstyle COS^- newcommandATANboxedmathttscriptstyle TAN^- abclist abc a: Gegenkathete von alpha c: Hypotenuse Rightarrow sinalpha * sinalpha frac ac && | c a c sinalpha . sin^circ .. * abc a: Ankathete von beta b: Gegenkathete von beta Rightarrow tanbeta * tanbeta frac ba && | a :tanbeta a fracbtanbeta frac.tan^circ .. * abc b: Ankathete von alpha c: Hypotenuse Rightarrow cosalpha * cosalpha frac bc && | c : cosalpha c fracbcosalpha frac.cos^circ .. * abc a: Ankathete von beta c: Hypotenuse Rightarrow cosbeta * cosbeta frac ac && | c a c cosbeta . cos^circ .. * abc a: Gegenkathete von alpha c: Hypotenuse Rightarrow sinalpha * sinalpha frac ac && | c:sinalpha c frac a sinalpha frac sin^circ . * abc b: Gegenkathete von beta c: Hypotenuse Rightarrow cosbeta * sinbeta frac bc && | c:sinbeta c fracbsinbeta frac.sin^circ .. * abc b: Ankathete von alpha c: Hypotenuse Rightarrow cosalpha * cosalpha frac bc && | c b c cosalpha . cos^circ .. * abc a: Ankathete von beta c: Hypotenuse Rightarrow cosbeta * cosbeta frac ac && | c :cosbeta c fraca cosbeta frac cos^circ . * abc a: Gegenkathete von alpha b: Ankathete von alpha Rightarrow tanalpha * tanalpha frac ab && | b :tanalpha b fracatanalpha frac.tan^circ .. * abc a: Ankathete von beta b: Gegenkathete von beta Rightarrow tanbeta * tanbeta frac ba && | a b a tanbeta . tan^circ .. * abc a: Gegenkathete von alpha b: Ankathete von alpha Rightarrow tanalpha * tanalpha frac ab && | b a b tanalpha . tan^circ .. * abc b: Gegenkathete von beta c: Hypotenuse Rightarrow sinbeta * sinbeta frac bc && | c b c sinbeta sin^circ . * abc a: Gegenkathete von alpha c: Hypotenuse Rightarrow sinalpha * sinalpha frac ac frac &&|AN alpha ^circ. * abc b: Ankathete von alpha c: Hypotenuse Rightarrow cosalpha * cosalpha frac bc frac.. &&|ACOS alpha .^circ. * abc a: Ankathete von beta b: Gegenkathete von beta Rightarrow tanbeta * tanbeta frac ba frac .. &&|ATAN beta .^circ. * abc a: Gegenkathete von alpha b: Ankathete von alpha Rightarrow tanalpha * tanalpha frac ab frac.. &&|ATAN alpha .^circ. * abc b: Gegenkathete von beta c: Hypotenuse Rightarrow sinbeta * sinbeta frac bc frac.. &&|AN beta .^circ. * abc a: Ankathete von beta c: Hypotenuse Rightarrow cosbeta * cosbeta frac ac frac.. &&|AN beta .^circ. * abclist
Meta Information
Exercise:
Betrachte das folge rechtwinklige Dreieck. center RechtwinkligesDreieckMitWinkel center Bestimme jeweils die gesuchte Grösse. Schreibe immer zuerst allgemein den Winkelsatz stelle ihn algebraisch nach der gesuchten Grösse um setze die gegebenen Grössen ein und berechne dann das Ergebnis! nprvmulticols abclist %%% newcommandgegges tabularrl Gegeben: & ##^circ ## Gesucht: & # tabular %%% abc geggesalphac.a abc geggesbetab.a abc geggesalphab.c abc geggesbetac.a abc geggesalphaac abc geggesbetab.c abc geggesalphac.b abc geggesbetaac abc geggesalphaa.b abc geggesbetaa.b abc geggesalphab.a abc geggesbetacb %%% renewcommandgegges tabularrl Gegeben: & ## ## Gesucht: & # tabular %%% abc gegges acalpha abc gegges b.c.alpha abc gegges a.b.beta abc gegges a.b.alpha abc gegges b.c.beta abc gegges a.c.beta abclist nprvmulticols
Solution:
newcommandANboxedmathttscriptstyle N^- newcommandACOSboxedmathttscriptstyle COS^- newcommandATANboxedmathttscriptstyle TAN^- abclist abc a: Gegenkathete von alpha c: Hypotenuse Rightarrow sinalpha * sinalpha frac ac && | c a c sinalpha . sin^circ .. * abc a: Ankathete von beta b: Gegenkathete von beta Rightarrow tanbeta * tanbeta frac ba && | a :tanbeta a fracbtanbeta frac.tan^circ .. * abc b: Ankathete von alpha c: Hypotenuse Rightarrow cosalpha * cosalpha frac bc && | c : cosalpha c fracbcosalpha frac.cos^circ .. * abc a: Ankathete von beta c: Hypotenuse Rightarrow cosbeta * cosbeta frac ac && | c a c cosbeta . cos^circ .. * abc a: Gegenkathete von alpha c: Hypotenuse Rightarrow sinalpha * sinalpha frac ac && | c:sinalpha c frac a sinalpha frac sin^circ . * abc b: Gegenkathete von beta c: Hypotenuse Rightarrow cosbeta * sinbeta frac bc && | c:sinbeta c fracbsinbeta frac.sin^circ .. * abc b: Ankathete von alpha c: Hypotenuse Rightarrow cosalpha * cosalpha frac bc && | c b c cosalpha . cos^circ .. * abc a: Ankathete von beta c: Hypotenuse Rightarrow cosbeta * cosbeta frac ac && | c :cosbeta c fraca cosbeta frac cos^circ . * abc a: Gegenkathete von alpha b: Ankathete von alpha Rightarrow tanalpha * tanalpha frac ab && | b :tanalpha b fracatanalpha frac.tan^circ .. * abc a: Ankathete von beta b: Gegenkathete von beta Rightarrow tanbeta * tanbeta frac ba && | a b a tanbeta . tan^circ .. * abc a: Gegenkathete von alpha b: Ankathete von alpha Rightarrow tanalpha * tanalpha frac ab && | b a b tanalpha . tan^circ .. * abc b: Gegenkathete von beta c: Hypotenuse Rightarrow sinbeta * sinbeta frac bc && | c b c sinbeta sin^circ . * abc a: Gegenkathete von alpha c: Hypotenuse Rightarrow sinalpha * sinalpha frac ac frac &&|AN alpha ^circ. * abc b: Ankathete von alpha c: Hypotenuse Rightarrow cosalpha * cosalpha frac bc frac.. &&|ACOS alpha .^circ. * abc a: Ankathete von beta b: Gegenkathete von beta Rightarrow tanbeta * tanbeta frac ba frac .. &&|ATAN beta .^circ. * abc a: Gegenkathete von alpha b: Ankathete von alpha Rightarrow tanalpha * tanalpha frac ab frac.. &&|ATAN alpha .^circ. * abc b: Gegenkathete von beta c: Hypotenuse Rightarrow sinbeta * sinbeta frac bc frac.. &&|AN beta .^circ. * abc a: Ankathete von beta c: Hypotenuse Rightarrow cosbeta * cosbeta frac ac frac.. &&|AN beta .^circ. * abclist
Betrachte das folge rechtwinklige Dreieck. center RechtwinkligesDreieckMitWinkel center Bestimme jeweils die gesuchte Grösse. Schreibe immer zuerst allgemein den Winkelsatz stelle ihn algebraisch nach der gesuchten Grösse um setze die gegebenen Grössen ein und berechne dann das Ergebnis! nprvmulticols abclist %%% newcommandgegges tabularrl Gegeben: & ##^circ ## Gesucht: & # tabular %%% abc geggesalphac.a abc geggesbetab.a abc geggesalphab.c abc geggesbetac.a abc geggesalphaac abc geggesbetab.c abc geggesalphac.b abc geggesbetaac abc geggesalphaa.b abc geggesbetaa.b abc geggesalphab.a abc geggesbetacb %%% renewcommandgegges tabularrl Gegeben: & ## ## Gesucht: & # tabular %%% abc gegges acalpha abc gegges b.c.alpha abc gegges a.b.beta abc gegges a.b.alpha abc gegges b.c.beta abc gegges a.c.beta abclist nprvmulticols
Solution:
newcommandANboxedmathttscriptstyle N^- newcommandACOSboxedmathttscriptstyle COS^- newcommandATANboxedmathttscriptstyle TAN^- abclist abc a: Gegenkathete von alpha c: Hypotenuse Rightarrow sinalpha * sinalpha frac ac && | c a c sinalpha . sin^circ .. * abc a: Ankathete von beta b: Gegenkathete von beta Rightarrow tanbeta * tanbeta frac ba && | a :tanbeta a fracbtanbeta frac.tan^circ .. * abc b: Ankathete von alpha c: Hypotenuse Rightarrow cosalpha * cosalpha frac bc && | c : cosalpha c fracbcosalpha frac.cos^circ .. * abc a: Ankathete von beta c: Hypotenuse Rightarrow cosbeta * cosbeta frac ac && | c a c cosbeta . cos^circ .. * abc a: Gegenkathete von alpha c: Hypotenuse Rightarrow sinalpha * sinalpha frac ac && | c:sinalpha c frac a sinalpha frac sin^circ . * abc b: Gegenkathete von beta c: Hypotenuse Rightarrow cosbeta * sinbeta frac bc && | c:sinbeta c fracbsinbeta frac.sin^circ .. * abc b: Ankathete von alpha c: Hypotenuse Rightarrow cosalpha * cosalpha frac bc && | c b c cosalpha . cos^circ .. * abc a: Ankathete von beta c: Hypotenuse Rightarrow cosbeta * cosbeta frac ac && | c :cosbeta c fraca cosbeta frac cos^circ . * abc a: Gegenkathete von alpha b: Ankathete von alpha Rightarrow tanalpha * tanalpha frac ab && | b :tanalpha b fracatanalpha frac.tan^circ .. * abc a: Ankathete von beta b: Gegenkathete von beta Rightarrow tanbeta * tanbeta frac ba && | a b a tanbeta . tan^circ .. * abc a: Gegenkathete von alpha b: Ankathete von alpha Rightarrow tanalpha * tanalpha frac ab && | b a b tanalpha . tan^circ .. * abc b: Gegenkathete von beta c: Hypotenuse Rightarrow sinbeta * sinbeta frac bc && | c b c sinbeta sin^circ . * abc a: Gegenkathete von alpha c: Hypotenuse Rightarrow sinalpha * sinalpha frac ac frac &&|AN alpha ^circ. * abc b: Ankathete von alpha c: Hypotenuse Rightarrow cosalpha * cosalpha frac bc frac.. &&|ACOS alpha .^circ. * abc a: Ankathete von beta b: Gegenkathete von beta Rightarrow tanbeta * tanbeta frac ba frac .. &&|ATAN beta .^circ. * abc a: Gegenkathete von alpha b: Ankathete von alpha Rightarrow tanalpha * tanalpha frac ab frac.. &&|ATAN alpha .^circ. * abc b: Gegenkathete von beta c: Hypotenuse Rightarrow sinbeta * sinbeta frac bc frac.. &&|AN beta .^circ. * abc a: Ankathete von beta c: Hypotenuse Rightarrow cosbeta * cosbeta frac ac frac.. &&|AN beta .^circ. * abclist
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Trigonometrie 1 by uz