Równanie trygonometryczne: Różnice pomiędzy wersjami
Z Nonsensopedii, polskiej encyklopedii humoru
M Znacznik: edytor źródłowy |
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(Nie pokazano 7 wersji utworzonych przez 6 użytkowników) | |||
Linia 1: | Linia 1: | ||
'''Równanie trygonometryczne''' – szczególny rodzaj [[Równanie|równania]], w którym oprócz literek i cyferek pojawiają się słówka ''sin'', ''cos'', ''tg'', ''ctg'', ''cosh'' itp.. Równania te zostały wynalezione przez starożytnych Greków w celu dręczenia dzieci objętych obowiązkiem edukacji matematyki. |
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==Sposób dowodzenia prawdziwości równania trygonometrycznego== |
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Aby dowieść prawdziwość równania trygonometrycznego, np.: |
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::''cos (x/n) = sin<sup>2</sup>(x) * π'' |
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należy: |
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* zapisać lewą i prawą stronę równania w odpowiednio lewy górnym i prawym dolnym rogu kartki |
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* korzystając z podstawowych praw [[Trygonometria|trygonometrii]] i [[Algebra|algebry]] w sposób dowolny przekształcać i rozwijać równania w następujący sposób: |
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**jeżeli rozwijamy lewą stronę równania, dopisujemy ''= nowe rozwinięcie'' po prawej stronie ostatniej wartości, lub linijkę niżej po lewej stronie |
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**jeżeli rozwijamy prawą stronę równania, dopisujemy ''nowe rozwinięcie ='' po lewej stronie ostatniej wartości, lub linijkę wyżej po prawej stronie |
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* gdy nie już miejsca na dopisanie nowej postaci, stawiamy ''='' łączący oba ciągi |
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===Przykład=== |
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Udowodnimy, że ''sin(x)=cos(x)'' dla przykładu niewielkiej kartki:<br/> |
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<math>sin(x) = sin(x) + 1 - 1</math><br/> |
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<math>cos(x)*(\sqrt{(cos(x)})^{(-2)} + cos(x) = cos(x)</math><br/> |
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<math>sin(x) = sin(x) + 1 - 1 = \sqrt{cos^2(x) - 1} + (\frac{3\sqrt3}{\sqrt{27}})</math><br/> |
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<math>cos(x)*(\sqrt{cos(x))^{-2} + cos(x)} = cos(x)</math> |
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<math>sin(x) = sin(x) + 1 - 1 = \sqrt{cos^2(x) - 1} + (\frac{3\sqrt3}{\sqrt{27}}) = cos(x)*(\sqrt{cos(x)^{(-2)}} + cos(x) = cos(x)</math> |
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W ten sposób udowodniliśmy, że ''sin(x)=cos(x)'' |
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{{stub|mat}} |
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{{Matematyka}} |
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[[Kategoria:Matematyka]] |
Aktualna wersja na dzień 23:26, 6 wrz 2017
Równanie trygonometryczne – szczególny rodzaj równania, w którym oprócz literek i cyferek pojawiają się słówka sin, cos, tg, ctg, cosh itp.. Równania te zostały wynalezione przez starożytnych Greków w celu dręczenia dzieci objętych obowiązkiem edukacji matematyki.
Sposób dowodzenia prawdziwości równania trygonometrycznego[edytuj • edytuj kod]
Aby dowieść prawdziwość równania trygonometrycznego, np.:
- cos (x/n) = sin2(x) * π
należy:
- zapisać lewą i prawą stronę równania w odpowiednio lewy górnym i prawym dolnym rogu kartki
- korzystając z podstawowych praw trygonometrii i algebry w sposób dowolny przekształcać i rozwijać równania w następujący sposób:
- jeżeli rozwijamy lewą stronę równania, dopisujemy = nowe rozwinięcie po prawej stronie ostatniej wartości, lub linijkę niżej po lewej stronie
- jeżeli rozwijamy prawą stronę równania, dopisujemy nowe rozwinięcie = po lewej stronie ostatniej wartości, lub linijkę wyżej po prawej stronie
- gdy nie już miejsca na dopisanie nowej postaci, stawiamy = łączący oba ciągi
Przykład[edytuj • edytuj kod]
Udowodnimy, że sin(x)=cos(x) dla przykładu niewielkiej kartki:
W ten sposób udowodniliśmy, że sin(x)=cos(x)
To jest tylko zalążek artykułu z dziedziny matematyki. Jeśli rozwiązywanie różniczek to twoje ulubione zajęcie – rozbuduj go.