Lösung 4.4:5c - Versionsgeschichte

Tek: Replaced figure with metapost figure

Tek — Tue, 25 Aug 2009 13:30:55 GMT

Replaced figure with metapost figure

← Nächstältere Version		Version vom 13:30, 25. Aug. 2009
Zeile 7:		Zeile 7:
	{{Abgesetzte Formel\|\|<math>v=u\qquad\text{und}\qquad v=-u\,\textrm{.}</math>}}		{{Abgesetzte Formel\|\|<math>v=u\qquad\text{und}\qquad v=-u\,\textrm{.}</math>}}

-	~~[[Image~~:~~4_4_5_c~~.~~gif\|~~center]]	+	<center>{{:4.4.5c - Solution - Two unit circles with angles u and -u, respectively}}</center>

	Also sind die allgemeinen Lösungen		Also sind die allgemeinen Lösungen

Markus.bez um 14:35, 19. Jun. 2009

Markus.bez — Fri, 19 Jun 2009 14:35:05 GMT

← Nächstältere Version		Version vom 14:35, 19. Jun. 2009
Zeile 1:		Zeile 1:
-	Für einen ~~Konstanten~~ ''u'', ist die Gleichung	+	Für einen Konstante ''u'' ist die Gleichung

	{{Abgesetzte Formel\|\|<math>\cos u=\cos v</math>}}		{{Abgesetzte Formel\|\|<math>\cos u=\cos v</math>}}

-	für folgende ~~Winkeln~~ ''v'' erfüllt	+	für folgende Winkel ''v'' erfüllt:

	{{Abgesetzte Formel\|\|<math>v=u\qquad\text{und}\qquad v=-u\,\textrm{.}</math>}}		{{Abgesetzte Formel\|\|<math>v=u\qquad\text{und}\qquad v=-u\,\textrm{.}</math>}}
Zeile 11:		Zeile 11:
	Also sind die allgemeinen Lösungen		Also sind die allgemeinen Lösungen

-	{{Abgesetzte Formel\|\|<math>v=u+2n\pi\qquad\text{und}\qquad v=-u+2n\pi\,,</math>}}	+	{{Abgesetzte Formel\|\|<math>v=u+2n\pi\qquad\text{und}\qquad v=-u+2n\pi\,.</math>}}

-	~~Also hat die~~ Gleichung	+	Die Gleichung

	{{Abgesetzte Formel\|\|<math>\cos 5x=\cos (x+\pi/5)</math>}}		{{Abgesetzte Formel\|\|<math>\cos 5x=\cos (x+\pi/5)</math>}}

-	die Lösungen	+	hat damit die Lösungen

	{{Abgesetzte Formel\|\|<math>\left\{\begin{align} 5x&=x+\frac{\pi}{5}+2n\pi\quad\text{oder}\\[5pt] 5x &= -x-\frac{\pi}{5}+2n\pi\,\textrm{.}\end{align}\right.</math>}}		{{Abgesetzte Formel\|\|<math>\left\{\begin{align} 5x&=x+\frac{\pi}{5}+2n\pi\quad\text{oder}\\[5pt] 5x &= -x-\frac{\pi}{5}+2n\pi\,\textrm{.}\end{align}\right.</math>}}

Martin um 11:21, 7. Apr. 2009

Martin — Tue, 07 Apr 2009 11:21:53 GMT

← Nächstältere Version		Version vom 11:21, 7. Apr. 2009
Zeile 1:		Zeile 1:
-	~~For a fixed value of~~ ''u'', ~~an equality of the form~~	+	Für einen Konstanten ''u'', ist die Gleichung

	{{Abgesetzte Formel\|\|<math>\cos u=\cos v</math>}}		{{Abgesetzte Formel\|\|<math>\cos u=\cos v</math>}}

-	~~is satisfied by two angles~~ ''v'' ~~in the unit circle,~~	+	für folgende Winkeln ''v'' erfüllt

-	{{Abgesetzte Formel\|\|<math>v=u\qquad\text{~~and~~}\qquad v=-u\,\textrm{.}</math>}}	+	{{Abgesetzte Formel\|\|<math>v=u\qquad\text{und}\qquad v=-u\,\textrm{.}</math>}}

	[[Image:4_4_5_c.gif\|center]]		[[Image:4_4_5_c.gif\|center]]

-	~~This means that all angles ''v'' which satisfy the equality are~~	+	Also sind die allgemeinen Lösungen

-	{{Abgesetzte Formel\|\|<math>v=u+2n\pi\qquad\text{~~and~~}\qquad v=-u+2n\pi\,,</math>}}	+	{{Abgesetzte Formel\|\|<math>v=u+2n\pi\qquad\text{und}\qquad v=-u+2n\pi\,,</math>}}

-	~~where ''n'' is an arbitrary integer.~~	+	Also hat die Gleichung
-		+
-	~~Therefore, the equation~~	+

	{{Abgesetzte Formel\|\|<math>\cos 5x=\cos (x+\pi/5)</math>}}		{{Abgesetzte Formel\|\|<math>\cos 5x=\cos (x+\pi/5)</math>}}

-	~~has the solutions~~	+	die Lösungen

-	{{Abgesetzte Formel\|\|<math>\left\{\begin{align} 5x&=x+\frac{\pi}{5}+2n\pi\quad\text{or}\\[5pt] 5x &= -x-\frac{\pi}{5}+2n\pi\,\textrm{.}\end{align}\right.</math>}}	+	{{Abgesetzte Formel\|\|<math>\left\{\begin{align} 5x&=x+\frac{\pi}{5}+2n\pi\quad\text{oder}\\[5pt] 5x &= -x-\frac{\pi}{5}+2n\pi\,\textrm{.}\end{align}\right.</math>}}

-	~~If we collect~~ ''x'' ~~onto one side~~, ~~we end up with~~	+	Lösen wir die Gleichung für ''x'', erhalten wir

	{{Abgesetzte Formel\|\|<math>\left\{\begin{align}		{{Abgesetzte Formel\|\|<math>\left\{\begin{align}
Zeile 29:		Zeile 27:
	x &= -\frac{\pi }{30}+\frac{n\pi}{3}\,,		x &= -\frac{\pi }{30}+\frac{n\pi}{3}\,,
	\end{align}\right.</math>}}		\end{align}\right.</math>}}
-
-	where ''n'' is an arbitrary integer.

Tekbot: hat „Solution 4.4:5c“ nach „Lösung 4.4:5c“ verschoben: Robot: moved page

Tekbot — Wed, 22 Oct 2008 15:16:44 GMT

hat „Solution 4.4:5c“ nach „Lösung 4.4:5c“ verschoben: Robot: moved page

← Nächstältere Version

Version vom 15:16, 22. Okt. 2008

Tekbot: Robot: Automated text replacement (-{{Displayed math +{{Abgesetzte Formel)

Tekbot — Wed, 22 Oct 2008 09:00:08 GMT

Robot: Automated text replacement (-{{Displayed math +{{Abgesetzte Formel)

← Nächstältere Version		Version vom 09:00, 22. Okt. 2008
Zeile 1:		Zeile 1:
	For a fixed value of ''u'', an equality of the form		For a fixed value of ''u'', an equality of the form

-	{{~~Displayed math~~\|\|<math>\cos u=\cos v</math>}}	+	{{Abgesetzte Formel\|\|<math>\cos u=\cos v</math>}}

	is satisfied by two angles ''v'' in the unit circle,		is satisfied by two angles ''v'' in the unit circle,

-	{{~~Displayed math~~\|\|<math>v=u\qquad\text{and}\qquad v=-u\,\textrm{.}</math>}}	+	{{Abgesetzte Formel\|\|<math>v=u\qquad\text{and}\qquad v=-u\,\textrm{.}</math>}}

	[[Image:4_4_5_c.gif\|center]]		[[Image:4_4_5_c.gif\|center]]
Zeile 11:		Zeile 11:
	This means that all angles ''v'' which satisfy the equality are		This means that all angles ''v'' which satisfy the equality are

-	{{~~Displayed math~~\|\|<math>v=u+2n\pi\qquad\text{and}\qquad v=-u+2n\pi\,,</math>}}	+	{{Abgesetzte Formel\|\|<math>v=u+2n\pi\qquad\text{and}\qquad v=-u+2n\pi\,,</math>}}

	where ''n'' is an arbitrary integer.		where ''n'' is an arbitrary integer.
Zeile 17:		Zeile 17:
	Therefore, the equation		Therefore, the equation

-	{{~~Displayed math~~\|\|<math>\cos 5x=\cos (x+\pi/5)</math>}}	+	{{Abgesetzte Formel\|\|<math>\cos 5x=\cos (x+\pi/5)</math>}}

	has the solutions		has the solutions

-	{{~~Displayed math~~\|\|<math>\left\{\begin{align} 5x&=x+\frac{\pi}{5}+2n\pi\quad\text{or}\\[5pt] 5x &= -x-\frac{\pi}{5}+2n\pi\,\textrm{.}\end{align}\right.</math>}}	+	{{Abgesetzte Formel\|\|<math>\left\{\begin{align} 5x&=x+\frac{\pi}{5}+2n\pi\quad\text{or}\\[5pt] 5x &= -x-\frac{\pi}{5}+2n\pi\,\textrm{.}\end{align}\right.</math>}}

	If we collect ''x'' onto one side, we end up with		If we collect ''x'' onto one side, we end up with

-	{{~~Displayed math~~\|\|<math>\left\{\begin{align}	+	{{Abgesetzte Formel\|\|<math>\left\{\begin{align}
	x &= \frac{\pi}{20} + \frac{n\pi}{2}\,,\\[5pt]		x &= \frac{\pi}{20} + \frac{n\pi}{2}\,,\\[5pt]
	x &= -\frac{\pi }{30}+\frac{n\pi}{3}\,,		x &= -\frac{\pi }{30}+\frac{n\pi}{3}\,,

Tek um 14:09, 13. Okt. 2008

Tek — Mon, 13 Oct 2008 14:09:51 GMT

← Nächstältere Version		Version vom 14:09, 13. Okt. 2008
Zeile 1:		Zeile 1:
-	For a fixed value of	+	For a fixed value of ''u'', an equality of the form
-	~~<math>~~u~~</math>~~, an equality of the form	+

		+	{{Displayed math\|\|<math>\cos u=\cos v</math>}}

-	~~<math>\cos u=\cos v</math>~~	+	is satisfied by two angles ''v'' in the unit circle,
-		+
-		+
-	is satisfied by two angles	+
-	~~<math>~~v~~</math>~~	+
-	in the unit circle:	+
-		+
-		+
-	~~<math>v=u</math>~~	+
-	~~and~~	+
-	~~<math>v=-u</math>~~	+

		+	{{Displayed math\|\|<math>v=u\qquad\text{and}\qquad v=-u\,\textrm{.}</math>}}

	[[Image:4_4_5_c.gif\|center]]		[[Image:4_4_5_c.gif\|center]]

-	This means that all angles	+	This means that all angles ''v'' which satisfy the equality are
-	~~<math>~~v~~</math>~~	+
-	which satisfy the equality are	+

		+	{{Displayed math\|\|<math>v=u+2n\pi\qquad\text{and}\qquad v=-u+2n\pi\,,</math>}}

-	~~<math>v=u+2n\pi </math>~~	+	where ''n'' is an arbitrary integer.
-	~~and~~	+
-	~~<math>v=-u+2n\pi </math>~~	+
-		+
-		+
-	where	+
-	~~<math>~~n~~\text{ }</math>~~	+
-	is an arbitrary integer.	+

	Therefore, the equation		Therefore, the equation

-		+	{{Displayed math\|\|<math>\cos 5x=\cos (x+\pi/5)</math>}}
-	<math>\cos 5x=\cos ~~\left~~( x+{\pi }/{5~~}\; \right~~)</math>	+
-		+

	has the solutions		has the solutions

-		+	{{Displayed math\|\|<math>\left\{\begin{align} 5x&=x+\frac{\pi}{5}+2n\pi\quad\text{or}\\[5pt] 5x &= -x-\frac{\pi}{5}+2n\pi\,\textrm{.}\end{align}\right.</math>}}
-	<math>5x=x+\frac{\pi }{5}+2n\pi ~~</math>~~	+
-	or	+
-		+
-	~~<math>~~5x=-x-\frac{\pi }{5}+2n\pi </math>	+

-	If we collect	+	If we collect ''x'' onto one side, we end up with
-	~~<math>~~x~~\text{ }</math>~~	+
-	onto one side, we end up with	+

		+	{{Displayed math\|\|<math>\left\{\begin{align}
		+	x &= \frac{\pi}{20} + \frac{n\pi}{2}\,,\\[5pt]
		+	x &= -\frac{\pi }{30}+\frac{n\pi}{3}\,,
		+	\end{align}\right.</math>}}

-	~~<math>\left\{ \begin{array}{*{35}l}~~	+	where ''n'' is an arbitrary integer.
-	~~x=\frac{\pi }{20}+\frac{1}{2}~~n~~\pi \\~~	+
-	~~x=-\frac{\pi }{30}+\frac{1}{3}n\pi \\~~	+
-	~~\end{array} \right.</math>~~	+
-	(	+
-	~~<math>n\text{ }</math>~~	+
-	an arbitrary integer).	+

Ian um 11:18, 1. Okt. 2008

Ian — Wed, 01 Oct 2008 11:18:33 GMT

← Nächstältere Version		Version vom 11:18, 1. Okt. 2008
Zeile 1:		Zeile 1:
-	~~{{NAVCONTENT_START}}~~	+	For a fixed value of
-	<~~center~~> ~~[[Image:4_4_5c-1(2).gif]]~~ </~~center~~>	+	<math>u</math>, an equality of the form
-	~~{{NAVCONTENT_STOP}}~~	+
-	~~{{NAVCONTENT_START}}~~	+
-	<~~center~~> ~~[[Image~~:~~4_4_5c-2(2).gif]]~~ </~~center~~>	+	<math>\cos u=\cos v</math>
-	~~{{NAVCONTENT_STOP}}~~	+
		+
		+	is satisfied by two angles
		+	<math>v</math>
		+	in the unit circle:
		+
		+
		+	<math>v=u</math>
		+	and
		+	<math>v=-u</math>


	[[Image:4_4_5_c.gif\|center]]		[[Image:4_4_5_c.gif\|center]]
		+
		+	This means that all angles
		+	<math>v</math>
		+	which satisfy the equality are
		+
		+
		+	<math>v=u+2n\pi </math>
		+	and
		+	<math>v=-u+2n\pi </math>
		+
		+
		+	where
		+	<math>n\text{ }</math>
		+	is an arbitrary integer.
		+
		+	Therefore, the equation
		+
		+
		+	<math>\cos 5x=\cos \left( x+{\pi }/{5}\; \right)</math>
		+
		+
		+	has the solutions
		+
		+
		+	<math>5x=x+\frac{\pi }{5}+2n\pi </math>
		+	or
		+
		+	<math>5x=-x-\frac{\pi }{5}+2n\pi </math>
		+
		+	If we collect
		+	<math>x\text{ }</math>
		+	onto one side, we end up with
		+
		+
		+	<math>\left\{ \begin{array}{*{35}l}
		+	x=\frac{\pi }{20}+\frac{1}{2}n\pi \\
		+	x=-\frac{\pi }{30}+\frac{1}{3}n\pi \\
		+	\end{array} \right.</math>
		+	(
		+	<math>n\text{ }</math>
		+	an arbitrary integer).

Tekbot: Lösning 4.4:5c moved to Solution 4.4:5c: Robot: moved page

Tekbot — Tue, 09 Sep 2008 10:06:23 GMT

Lösning 4.4:5c moved to Solution 4.4:5c: Robot: moved page

← Nächstältere Version

Version vom 10:06, 9. Sep. 2008

Tekbot: Robot: Automated text replacement (-[[Bild: +[[Image:)

Tekbot — Thu, 21 Aug 2008 07:20:19 GMT

Robot: Automated text replacement (-[[Bild: +[[Image:)

← Nächstältere Version		Version vom 07:20, 21. Aug. 2008
Zeile 1:		Zeile 1:
	{{NAVCONTENT_START}}		{{NAVCONTENT_START}}
-	<center> [[~~Bild~~:4_4_5c-1(2).gif]] </center>	+	<center> [[Image:4_4_5c-1(2).gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}
	{{NAVCONTENT_START}}		{{NAVCONTENT_START}}
-	<center> [[~~Bild~~:4_4_5c-2(2).gif]] </center>	+	<center> [[Image:4_4_5c-2(2).gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}


-	[[~~Bild~~:4_4_5_c.gif\|center]]	+	[[Image:4_4_5_c.gif\|center]]

Madde um 13:12, 19. Aug. 2008

Madde — Tue, 19 Aug 2008 13:12:17 GMT

← Nächstältere Version		Version vom 13:12, 19. Aug. 2008
Zeile 5:		Zeile 5:
	<center> [[Bild:4_4_5c-2(2).gif]] </center>		<center> [[Bild:4_4_5c-2(2).gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}
		+
		+
		+	[[Bild:4_4_5_c.gif\|center]]