Równanie trygonometryczne: Różnice pomiędzy wersjami
M (Przywrócono przedostatnią wersję, jej autor to Ełek. Autor wycofanej wersji to 83.19.211.122.) |
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'''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 lewą 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: |
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<pre> |
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sin(x) |
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cos(x) |
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</pre> |
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<pre> |
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sin(x) = sin(x) + 1 - 1 |
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cos(x)*(sqrt(cos(x))^(-2) + cos(x) = cos(x) |
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</pre> |
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<pre> |
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sin(x) = sin(x) + 1 - 1 = sqrt(cos^2(x) - 1) + (3sqrt(3)/sqrt(27)) |
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cos(x)*(sqrt(cos(x))^(-2) + cos(x) = cos(x) |
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</pre> |
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<pre> |
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sin(x) = sin(x) + 1 - 1 = sqrt(cos^2(x) - 1) + (3sqrt(3)/sqrt(27)) = |
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= cos(x)*(sqrt(cos(x))^(-2) + cos(x) = cos(x) |
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</pre> |
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W ten sposób udowodniliśmy, że ''sin(x)=cos(x)'' |
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{{matematyka stub}} |
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[[Kategoria:Matematyka]] |
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Wersja z 15:24, 30 mar 2009
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