Równanie trygonometryczne: Różnice pomiędzy wersjami

Z Nonsensopedii, polskiej encyklopedii humoru
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Linia 1: Linia 1:
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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.

==Sposób dowodzenia prawdziwości równania trygonometrycznego==
Aby dowieść prawdziwość równania trygonometrycznego, np.:
::''cos (x/n) = sin<sup>2</sup>(x) * &pi;''

należy:

* zapisać lewą i prawą stronę równania w odpowiednio lewy górnym i prawym dolnym rogu kartki
* korzystając z podstawowych praw [[Trygonometria|trygonometrii]] i [[Algebra|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 lewą 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===
Udowodnimy, że ''sin(x)=cos(x)'' dla przykładu niewielkiej kartki:

<pre>
sin(x)

cos(x)
</pre>

<pre>
sin(x) = sin(x) + 1 - 1

cos(x)*(sqrt(cos(x))^(-2) + cos(x) = cos(x)
</pre>

<pre>
sin(x) = sin(x) + 1 - 1 = sqrt(cos^2(x) - 1) + (3sqrt(3)/sqrt(27))

cos(x)*(sqrt(cos(x))^(-2) + cos(x) = cos(x)
</pre>

<pre>
sin(x) = sin(x) + 1 - 1 = sqrt(cos^2(x) - 1) + (3sqrt(3)/sqrt(27)) =

= cos(x)*(sqrt(cos(x))^(-2) + cos(x) = cos(x)
</pre>

W ten sposób udowodniliśmy, że ''sin(x)=cos(x)''

{{matematyka stub}}

[[Kategoria:Matematyka]]

Wersja z 12:47, 30 mar 2009

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