Lösung 4.4:5a - Versionsgeschichte

Tek: Replaced figure with metapost figure

Tek — Tue, 25 Aug 2009 13:22:14 GMT

Replaced figure with metapost figure

← Nächstältere Version		Version vom 13:22, 25. Aug. 2009
Zeile 7:		Zeile 7:
	{{Abgesetzte Formel\|\|<math>v=u\qquad\text{und}\qquad v=\pi-u\,\textrm{.}</math>}}		{{Abgesetzte Formel\|\|<math>v=u\qquad\text{und}\qquad v=\pi-u\,\textrm{.}</math>}}

-	~~[[Image~~:~~4_4_5_a~~.~~gif\|\|~~center]]	+	<center>{{:4.4.5a - Solution - Two unit circles with angles v = u and v = π - u}}</center>

	(Die einzige Ausnahme ist, wenn <math>u = \pi/2</math> oder <math>u=3\pi/2</math>, da in diesen Fällen <math>u</math> und <math>\pi-u</math> dieselben Winkel sind)		(Die einzige Ausnahme ist, wenn <math>u = \pi/2</math> oder <math>u=3\pi/2</math>, da in diesen Fällen <math>u</math> und <math>\pi-u</math> dieselben Winkel sind)

Markus.bez um 14:32, 19. Jun. 2009

Markus.bez — Fri, 19 Jun 2009 14:32:58 GMT

← Nächstältere Version		Version vom 14:32, 19. Jun. 2009
Zeile 1:		Zeile 1:
	Betrachten wir die Gleichung		Betrachten wir die Gleichung

-	{{Abgesetzte Formel\|\|<math>\sin u = \sin v</math>\|(*)}}	+	{{Abgesetzte Formel\|\|<math>\sin u = \sin v,</math>\|(*)}}

-	wo ''u'' ~~ein Konstant~~ ist, gibt es zwei Lösungen für diese Gleichung, nämlich	+	wobei ''u'' eine Konstante ist, gibt es zwei Lösungen für diese Gleichung, nämlich

	{{Abgesetzte Formel\|\|<math>v=u\qquad\text{und}\qquad v=\pi-u\,\textrm{.}</math>}}		{{Abgesetzte Formel\|\|<math>v=u\qquad\text{und}\qquad v=\pi-u\,\textrm{.}</math>}}
Zeile 9:		Zeile 9:
	[[Image:4_4_5_a.gif\|\|center]]		[[Image:4_4_5_a.gif\|\|center]]

-	(Die einzige Ausnahme ist wenn <math>u = \pi/2</math> oder <math>u=3\pi/2</math>, da in diesen Fällen <math>u</math> und <math>\pi-u</math> dieselben ~~Winkeln~~ sind)	+	(Die einzige Ausnahme ist, wenn <math>u = \pi/2</math> oder <math>u=3\pi/2</math>, da in diesen Fällen <math>u</math> und <math>\pi-u</math> dieselben Winkel sind)

-	Wir erhalten die ~~algemeine~~ Lösung indem wir ~~einen Multipel~~ von <math>2\pi</math> zur Lösung addieren,	+	Wir erhalten die allgemeine Lösung, indem wir ein Vielfaches von <math>2\pi</math> zur Lösung addieren:

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

	Für unsere Gleichung		Für unsere Gleichung

Martin um 10:53, 7. Apr. 2009

Martin — Tue, 07 Apr 2009 10:53:23 GMT

← Nächstältere Version		Version vom 10:53, 7. Apr. 2009
Zeile 1:		Zeile 1:
-	~~If we consider for a moment the equality~~	+	Betrachten wir die Gleichung

	{{Abgesetzte Formel\|\|<math>\sin u = \sin v</math>\|(*)}}		{{Abgesetzte Formel\|\|<math>\sin u = \sin v</math>\|(*)}}

-	~~where~~ ''u'' ~~has a fixed value~~, ~~there are usually two angles ''v'' in the unit circle which ensure that the equality holds~~,	+	wo ''u'' ein Konstant ist, gibt es zwei Lösungen für diese Gleichung, nämlich

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

	[[Image:4_4_5_a.gif\|\|center]]		[[Image:4_4_5_a.gif\|\|center]]

-	(~~The only exception is when~~ <math>u = \pi/2</math> or <math>u=3\pi/2</math>, in ~~which case~~ <math>u</math> ~~and~~ <math>\pi-u</math> ~~correspond to the same direction and there is only one angle ''v'' which satisfies the equality.~~)	+	(Die einzige Ausnahme ist wenn <math>u = \pi/2</math> oder <math>u=3\pi/2</math>, da in diesen Fällen <math>u</math> und <math>\pi-u</math> dieselben Winkeln sind)

-	~~We obtain all the angles ''v'' which satisfy (*) by adding multiples of~~ <math>2\pi</math>,	+	Wir erhalten die algemeine Lösung indem wir einen Multipel von <math>2\pi</math> zur Lösung addieren,

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

-	~~where ''n'' is an arbitrary integer.~~	+	Für unsere Gleichung
-		+
-	~~If we now go back to our equation~~	+

	{{Abgesetzte Formel\|\|<math>\sin 3x = \sin x</math>}}		{{Abgesetzte Formel\|\|<math>\sin 3x = \sin x</math>}}

-	~~the reasoning above shows that the equation is only satisfied when~~	+	erhalten wir die Lösungen

-	{{Abgesetzte Formel\|\|<math>3x = x+2n\pi\qquad\text{or}\qquad 3x = \pi-x+2n\pi\,\textrm{.}</math>}}	+	{{Abgesetzte Formel\|\|<math>3x = x+2n\pi\qquad\text{oder}\qquad 3x = \pi-x+2n\pi\,\textrm{.}</math>}}

-	~~If we make~~ ''x'' ~~the subject of each equation, we obtain the full solution to the equation~~,	+	Lösen wir ''x'', erhalten wir

	{{Abgesetzte Formel\|\|<math>\left\{\begin{align}		{{Abgesetzte Formel\|\|<math>\left\{\begin{align}

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

Tekbot — Wed, 22 Oct 2008 15:16:04 GMT

hat „Solution 4.4:5a“ nach „Lösung 4.4:5a“ 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 08:59:48 GMT

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

← Nächstältere Version		Version vom 08:59, 22. Okt. 2008
Zeile 1:		Zeile 1:
	If we consider for a moment the equality		If we consider for a moment the equality

-	{{~~Displayed math~~\|\|<math>\sin u = \sin v</math>\|(*)}}	+	{{Abgesetzte Formel\|\|<math>\sin u = \sin v</math>\|(*)}}

	where ''u'' has a fixed value, there are usually two angles ''v'' in the unit circle which ensure that the equality holds,		where ''u'' has a fixed value, there are usually two angles ''v'' in the unit circle which ensure that the equality holds,

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

	[[Image:4_4_5_a.gif\|\|center]]		[[Image:4_4_5_a.gif\|\|center]]
Zeile 13:		Zeile 13:
	We obtain all the angles ''v'' which satisfy (*) by adding multiples of <math>2\pi</math>,		We obtain all the angles ''v'' which satisfy (*) by adding multiples of <math>2\pi</math>,

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

	where ''n'' is an arbitrary integer.		where ''n'' is an arbitrary integer.
Zeile 19:		Zeile 19:
	If we now go back to our equation		If we now go back to our equation

-	{{~~Displayed math~~\|\|<math>\sin 3x = \sin x</math>}}	+	{{Abgesetzte Formel\|\|<math>\sin 3x = \sin x</math>}}

	the reasoning above shows that the equation is only satisfied when		the reasoning above shows that the equation is only satisfied when

-	{{~~Displayed math~~\|\|<math>3x = x+2n\pi\qquad\text{or}\qquad 3x = \pi-x+2n\pi\,\textrm{.}</math>}}	+	{{Abgesetzte Formel\|\|<math>3x = x+2n\pi\qquad\text{or}\qquad 3x = \pi-x+2n\pi\,\textrm{.}</math>}}

	If we make ''x'' the subject of each equation, we obtain the full solution to the equation,		If we make ''x'' the subject of each equation, we obtain the full solution to the equation,

-	{{~~Displayed math~~\|\|<math>\left\{\begin{align}	+	{{Abgesetzte Formel\|\|<math>\left\{\begin{align}
	x &= 0+n\pi\,,\\[5pt]		x &= 0+n\pi\,,\\[5pt]
	x &= \frac{\pi}{4}+\frac{n\pi}{2}\,\textrm{.}		x &= \frac{\pi}{4}+\frac{n\pi}{2}\,\textrm{.}
	\end{align}\right.</math>}}		\end{align}\right.</math>}}

Tek um 13:53, 13. Okt. 2008

Tek — Mon, 13 Oct 2008 13:53:18 GMT

← Nächstältere Version		Version vom 13:53, 13. Okt. 2008
Zeile 1:		Zeile 1:
	If we consider for a moment the equality		If we consider for a moment the equality

		+	{{Displayed math\|\|<math>\sin u = \sin v</math>\|(*)}}

-	~~<math>\sin~~ u~~=\sin~~ v~~\quad \quad \quad (*)</math>~~	+	where ''u'' has a fixed value, there are usually two angles ''v'' in the unit circle which ensure that the equality holds,

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

		+	[[Image:4_4_5_a.gif\|\|center]]

-	~~where~~	+	(The only exception is when <math>u = \pi/2</math> or <math>u=3\pi/2</math>, in which case <math>u</math> and <math>\pi-u</math> correspond to the same direction and there is only one angle ''v'' which satisfies the equality.)
-	<math>u</math>	+
-	~~has a fixed value, there are usually two angles~~	+
-	<math>v</math>	+
-	in the ~~unit circle~~ which ~~ensure that~~ the equality ~~holds:~~	+

		+	We obtain all the angles ''v'' which satisfy (*) by adding multiples of <math>2\pi</math>,

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

-		+	where ''n'' is an arbitrary integer.
-	~~[[Image:4_4_5_a.gif]]~~	+
-		+
-		+
-	~~(The only exception is when~~	+
-	~~<math>u={\pi }/{2}\;\text{ }</math>~~	+
-	or	+
-	~~<math>u=3{\pi }/{2}\;\text{ }</math>, in which case~~	+
-	~~<math>u</math>~~	+
-	~~and~~	+
-	~~<math>\pi -u\text{ }</math>~~	+
-	~~correspond to the same direction and there is only one angle~~	+
-	~~<math>v</math>~~	+
-	~~which satisfies the equality.)~~	+
-		+
-	~~We obtain all the angles~~	+
-	~~<math>v</math>~~	+
-	~~which satisfy (*) by adding multiples of~~	+
-	~~<math>\text{2}\pi </math>,~~	+
-		+
-		+
-	~~<math>v=u+2n\pi </math>~~	+
-	~~and~~	+
-	~~<math>v=\pi -u+2n\pi </math>~~	+
-		+
-		+
-	where	+
-	~~<math>~~n~~\text{ }</math>~~	+
-	is an arbitrary integer.	+

	If we now go back to our equation		If we now go back to our equation

		+	{{Displayed math\|\|<math>\sin 3x = \sin x</math>}}

-	~~<math>\sin 3x=\sin x</math>~~	+	the reasoning above shows that the equation is only satisfied when
-		+
-		+
-	the reasoning above shows that the equation is only satisfied when	+
-		+
-		+
-	~~<math>3x=x+2n\pi </math>~~	+
-	or	+
-	~~<math>3x=\pi -x+2n\pi </math>~~	+
-		+

-	~~If we make~~	+	{{Displayed math\|\|<math>3x = x+2n\pi\qquad\text{or}\qquad 3x = \pi-x+2n\pi\,\textrm{.}</math>}}
-	<math>x</math>	+
-	~~the subject of each equation, we obtain the full solution to the equation:~~	+

		+	If we make ''x'' the subject of each equation, we obtain the full solution to the equation,

-	<math>\left\{ \begin{~~array}{*{35}l~~}	+	{{Displayed math\|\|<math>\left\{\begin{align}
-	x=0+n\pi \\	+	x &= 0+n\pi\,,\\[5pt]
-	x=\frac{\pi }{4}+\frac{1}{2}~~n\pi~~ \\	+	x &= \frac{\pi}{4}+\frac{n\pi}{2}\,\textrm{.}
-	\end{~~array~~} \right.</math>	+	\end{align}\right.</math>}}

Ian um 10:55, 1. Okt. 2008

Ian — Wed, 01 Oct 2008 10:55:48 GMT

← Nächstältere Version		Version vom 10:55, 1. Okt. 2008
Zeile 1:		Zeile 1:
-	~~{{NAVCONTENT_START}}~~	+	If we consider for a moment the equality
-	<~~center~~> ~~[[Image:4_4_5a-1~~(2)~~.gif]]~~ </~~center~~>	+
-	~~{{NAVCONTENT_STOP}}~~	+
-	~~{{NAVCONTENT_START}}~~	+	<math>\sin u=\sin v\quad \quad \quad (*)</math>
-	<~~center~~> ~~[[Image~~:~~4_4_5a~~-~~2(2).gif]]~~ </~~center~~>	+
-	~~{{NAVCONTENT_STOP}}~~	+
		+
		+	where
		+	<math>u</math>
		+	has a fixed value, there are usually two angles
		+	<math>v</math>
		+	in the unit circle which ensure that the equality holds:
		+
		+
		+	<math>v=u</math>
		+	and
		+	<math>v=\pi -u</math>
		+

	[[Image:4_4_5_a.gif]]		[[Image:4_4_5_a.gif]]
		+
		+
		+	(The only exception is when
		+	<math>u={\pi }/{2}\;\text{ }</math>
		+	or
		+	<math>u=3{\pi }/{2}\;\text{ }</math>, in which case
		+	<math>u</math>
		+	and
		+	<math>\pi -u\text{ }</math>
		+	correspond to the same direction and there is only one angle
		+	<math>v</math>
		+	which satisfies the equality.)
		+
		+	We obtain all the angles
		+	<math>v</math>
		+	which satisfy (*) by adding multiples of
		+	<math>\text{2}\pi </math>,
		+
		+
		+	<math>v=u+2n\pi </math>
		+	and
		+	<math>v=\pi -u+2n\pi </math>
		+
		+
		+	where
		+	<math>n\text{ }</math>
		+	is an arbitrary integer.
		+
		+	If we now go back to our equation
		+
		+
		+	<math>\sin 3x=\sin x</math>
		+
		+
		+	the reasoning above shows that the equation is only satisfied when
		+
		+
		+	<math>3x=x+2n\pi </math>
		+	or
		+	<math>3x=\pi -x+2n\pi </math>
		+
		+
		+	If we make
		+	<math>x</math>
		+	the subject of each equation, we obtain the full solution to the equation:
		+
		+
		+	<math>\left\{ \begin{array}{*{35}l}
		+	x=0+n\pi \\
		+	x=\frac{\pi }{4}+\frac{1}{2}n\pi \\
		+	\end{array} \right.</math>

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

Tekbot — Tue, 09 Sep 2008 10:05:43 GMT

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

← Nächstältere Version

Version vom 10:05, 9. Sep. 2008

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

Tekbot — Thu, 21 Aug 2008 07:19:59 GMT

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

← Nächstältere Version		Version vom 07:19, 21. Aug. 2008
Zeile 1:		Zeile 1:
	{{NAVCONTENT_START}}		{{NAVCONTENT_START}}
-	<center> [[~~Bild~~:4_4_5a-1(2).gif]] </center>	+	<center> [[Image:4_4_5a-1(2).gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}
	{{NAVCONTENT_START}}		{{NAVCONTENT_START}}
-	<center> [[~~Bild~~:4_4_5a-2(2).gif]] </center>	+	<center> [[Image:4_4_5a-2(2).gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}

-	[[~~Bild~~:4_4_5_a.gif]]	+	[[Image:4_4_5_a.gif]]

Madde um 13:10, 19. Aug. 2008

Madde — Tue, 19 Aug 2008 13:10:15 GMT

← Nächstältere Version		Version vom 13:10, 19. Aug. 2008
Zeile 5:		Zeile 5:
	<center> [[Bild:4_4_5a-2(2).gif]] </center>		<center> [[Bild:4_4_5a-2(2).gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}
		+
		+	[[Bild:4_4_5_a.gif]]