Lösung 4.4:7c - Versionsgeschichte

Tek: Replaced figures with metapost figures

2009-08-25T13:36:47Z

Replaced figures with metapost figures

← Nächstältere Version		Version vom 13:36, 25. Aug. 2009
Zeile 5:		Zeile 5:
	Wir wissen, dass es für ein fixes ''u'' zwei Winkel auf dem Einheitskreis gibt, für welche die Gleichung erfüllt ist, nämlich <math>v=u</math> und <math>v=-u</math>.		Wir wissen, dass es für ein fixes ''u'' zwei Winkel auf dem Einheitskreis gibt, für welche die Gleichung erfüllt ist, nämlich <math>v=u</math> und <math>v=-u</math>.

-	~~[[Image~~:~~4_4_7_c1~~.~~gif\|~~center]]	+	<center>{{:4.4.7c - Solution - Two unit circles with angles u and -u, respectively}}</center>

	Jetzt drehen wir auf dem Einheitskreis den Winkel <math>\pi/2</math> gegen den Uhrzeigersinn. Die Gerade <math>x=\cos u</math> wird dann <math>y=\cos u</math> und die Winkel ''u'' und -''u'' werden <math>u+\pi/2</math> und <math>-u+\pi/2</math>.		Jetzt drehen wir auf dem Einheitskreis den Winkel <math>\pi/2</math> gegen den Uhrzeigersinn. Die Gerade <math>x=\cos u</math> wird dann <math>y=\cos u</math> und die Winkel ''u'' und -''u'' werden <math>u+\pi/2</math> und <math>-u+\pi/2</math>.

-	~~[[Image~~:~~4_4_7_c2~~.~~gif\|~~center]]	+	<center>{{:4.4.7c - Solution - Two unit circles with angles u + π/2 and -u + π/2, respectively}}</center>

	Die Winkel <math>u+\pi/2</math> und <math>-u+\pi/2</math> haben daher dieselbe ''y''-Koordinate und daher denselben Sinus. Also ist die Gleichung		Die Winkel <math>u+\pi/2</math> und <math>-u+\pi/2</math> haben daher dieselbe ''y''-Koordinate und daher denselben Sinus. Also ist die Gleichung

Markus.bez um 15:01, 19. Jun. 2009

2009-06-19T15:01:14Z

Martin um 12:06, 7. Apr. 2009

2009-04-07T12:06:47Z

← Nächstältere Version		Version vom 12:06, 7. Apr. 2009
Zeile 1:		Zeile 1:
-	~~If we want to solve the equation~~ <math>\cos 3x = \sin 4x</math>, ~~we need an additional result which tells us for which values of ''u'' and ''v'' the equality~~	+	Um die Gleichung <math>\cos 3x = \sin 4x</math> zu lösen, betrachten wir zuerst die allgemeine Gleichung
-	~~<math>\cos u = \sin v</math> holds, but to get that we have to start with the equality <math>\cos u=\cos v</math>.~~	+
-		+
-	~~So, we start by looking at the equality~~	+

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

-	~~We know that for fixed~~ ''u'' ~~there are two angles~~ <math>v=u</math> ~~and~~ <math>v=-u~~</math> in the unit circle which have the cosine value <math>\cos u</math>, i.e. their ''x''-coordinate is equal to <math>\cos u\,~~</math>.	+	Wir wissen dass es für ein fixes ''u'' zwei Winkeln im Einheitskreis gibt für welche die Gleichung erfüllt ist, nämlich <math>v=u</math> und <math>v=-u</math>.

	[[Image:4_4_7_c1.gif\|center]]		[[Image:4_4_7_c1.gif\|center]]

-	~~Imagine now that the whole unit circle is rotated anti-clockwise an angle~~ <math>\pi/2</math>. ~~The line~~ <math>x=\cos u</math> ~~will become the line~~ <math>y=\cos u</math> ~~and the angles~~ ''u'' ~~and~~ -''u'' ~~are rotated to~~ <math>u+\pi/2</math> ~~and~~ <math>-u+\pi/2</math>, ~~respectively.~~	+	Jetzt drehen wir den Einheitskreis den Winkel <math>\pi/2</math> gegen den Uhrzeiger. Die Gerade <math>x=\cos u</math> bekommt dann <math>y=\cos u</math> und die Winkeln ''u'' und -''u'' bekommen <math>u+\pi/2</math> und <math>-u+\pi/2</math>,

	[[Image:4_4_7_c2.gif\|center]]		[[Image:4_4_7_c2.gif\|center]]

-	~~The angles~~ <math>u+\pi/2</math> ~~and~~ <math>-u+\pi/2</math> ~~therefore have their~~ ''y''-~~coordinate~~, ~~and hence sine value, equal to <math>\cos u</math>~~. ~~In other words, the equality~~	+	Die Winkeln <math>u+\pi/2</math> und <math>-u+\pi/2</math> haben daher dieselbe ''y''-Koordinate, und daher denselben Sinus. Also ist die Gleichung

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

-	~~holds for fixed~~ ''u'' ~~in the unit circle when~~ <math>v = \pm u + \pi/2</math>, ~~and more generally when~~	+	für ein fixes ''u'' erfüllt wenn <math>v = \pm u + \pi/2</math>, und allgemein wenn

-	{{Abgesetzte Formel\|\|<math>v = \pm u + \frac{\pi}{2} + 2n\pi\qquad</math>~~(''n'' is an arbitrary integer).~~}}	+	{{Abgesetzte Formel\|\|<math>v = \pm u + \frac{\pi}{2} + 2n\pi\qquad</math>}}

-	~~For our equation~~ <math>\cos 3x = \sin 4x</math>~~, this result means that ''x'' must satisfy~~	+	In unseren Fall ist die Gleichung <math>\cos 3x = \sin 4x</math> erfüllt wenn

	{{Abgesetzte Formel\|\|<math>4x = \pm 3x + \frac{\pi}{2} + 2n\pi\,\textrm{.}</math>}}		{{Abgesetzte Formel\|\|<math>4x = \pm 3x + \frac{\pi}{2} + 2n\pi\,\textrm{.}</math>}}

-	~~This means that the solutions to the equation are~~	+	und also bekommen wir die allgemeinen Lösungen

	{{Abgesetzte Formel\|\|<math>\left\{\begin{align}		{{Abgesetzte Formel\|\|<math>\left\{\begin{align}
Zeile 32:		Zeile 29:
	x &= \frac{\pi}{14} + \frac{2}{7}\pi n\,,		x &= \frac{\pi}{14} + \frac{2}{7}\pi n\,,
	\end{align}\right.</math>}}		\end{align}\right.</math>}}
-
-	where ''n'' is an arbitrary integer.

Tekbot: hat „Solution 4.4:7c“ nach „Lösung 4.4:7c“ verschoben: Robot: moved page

2008-10-22T15:18:44Z

hat „Solution 4.4:7c“ nach „Lösung 4.4:7c“ verschoben: Robot: moved page

← Nächstältere Version

Version vom 15:18, 22. Okt. 2008

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

2008-10-22T09:01:08Z

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

Tek um 07:59, 14. Okt. 2008

2008-10-14T07:59:06Z

← Nächstältere Version		Version vom 07:59, 14. Okt. 2008
Zeile 1:		Zeile 1:
-	If we want to solve the equation	+	If we want to solve the equation <math>\cos 3x = \sin 4x</math>, we need an additional result which tells us for which values of ''u'' and ''v'' the equality
-	<math>\~~text{~~cos ~~3}x~~=\~~text{~~sin ~~4}x~~</math>, we need an additional result which tells us for which values of	+	<math>\cos u = \sin v</math> holds, but to get that we have to start with the equality <math>\cos u=\cos v</math>.
-	~~<math>~~u~~</math>~~	+
-	and	+
-	~~<math>~~v~~</math>~~	+
-	the equality	+
-	<math>\~~text{~~cos }u=\~~text{~~sin }v</math>	+
-	holds, but to get that we have to start with the equality	+
-	<math>\cos u=\cos v</math>.	+

	So, we start by looking at the equality		So, we start by looking at the equality

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

-	~~<math>\cos u=\cos v</math>~~	+	We know that for fixed ''u'' there are two angles <math>v=u</math> and <math>v=-u</math> in the unit circle which have the cosine value <math>\cos u</math>, i.e. their ''x''-coordinate is equal to <math>\cos u\,</math>.
-		+
-		+
-	We know that for fixed	+
-	~~<math>~~u~~</math>~~	+
-	there are two angles	+
-	<math>v=u~~\text{ }~~</math>	+
-	and	+
-	<math>v=-~~\text{~~u}</math>	+
-	in the unit circle which have the cosine value	+
-	<math>\cos u</math>, i.e. their	+
-	~~<math>~~x~~</math>~~	+
-	-coordinate is equal to	+
-	<math>\cos u</math>.	+
-		+

	[[Image:4_4_7_c1.gif\|center]]		[[Image:4_4_7_c1.gif\|center]]

-	Imagine now that the whole unit circle is rotated anti-clockwise an angle	+	Imagine now that the whole unit circle is rotated anti-clockwise an angle <math>\pi/2</math>. The line <math>x=\cos u</math> will become the line <math>y=\cos u</math> and the angles ''u'' and -''u'' are rotated to <math>u+\pi/2</math> and <math>-u+\pi/2</math>, respectively.
-	<math>{\pi }/{2~~}\;~~</math>. The line	+
-	<math>x=\cos u</math>	+
-	will become the line	+
-	<math>y=\cos u</math>	+
-	and the angles	+
-	~~<math>~~u~~</math>~~	+
-	and	+
-	~~<math>~~-u~~</math>~~	+
-	are rotated to	+
-	<math>u+{\pi }/{2~~}\;~~</math>	+
-	and	+
-	<math>-u+{\pi }/{2~~}\;~~</math>, respectively.	+
-		+

	[[Image:4_4_7_c2.gif\|center]]		[[Image:4_4_7_c2.gif\|center]]

-	The angles	+	The angles <math>u+\pi/2</math> and <math>-u+\pi/2</math> therefore have their ''y''-coordinate, and hence sine value, equal to <math>\cos u</math>. In other words, the equality
-	<math>u+{\pi }/{2~~}\;~~</math>	+
-	and	+
-	<math>-u+{\pi }/{2~~}\;~~</math>	+
-	therefore have their	+
-	~~<math>~~y~~</math>~~	+
-	-coordinate, and hence sine value, equal to	+
-	<math>\cos u</math>. In other words, the equality	+
-		+
-		+
-	~~<math>\text{cos }u=\text{sin }v</math>~~	+
-		+
-		+
-	~~holds for fixed~~	+
-	~~<math>u</math>~~	+
-	~~in the unit circle when~~	+
-	~~<math>v=\pm u+{\pi }/{2}\;</math>, and more generally when~~	+
-		+

-	<math>v=\pm u+\~~frac{\pi }{2}+2n\pi~~ </math>	+	{{Displayed math\|\|<math>\cos u = \sin v</math>}}
-	(	+
-	~~<math>n</math>~~	+
-	~~an arbitrary integer).~~	+

-	~~For our equation~~	+	holds for fixed ''u'' in the unit circle when <math>v = \pm u + \pi/2</math>, and more generally when
-	<math>~~\text{cos 3}x~~=\~~text{sin 4}x~~</math>, ~~this result means that~~	+
-	~~<math>x\text{ }</math>~~	+
-	~~must satisfy~~	+

		+	{{Displayed math\|\|<math>v = \pm u + \frac{\pi}{2} + 2n\pi\qquad</math>(''n'' is an arbitrary integer).}}

-	<math>~~4x=~~\pm 3x+\~~frac{\pi }{2}+2n\pi~~ </math>	+	For our equation <math>\cos 3x = \sin 4x</math>, this result means that ''x'' must satisfy

		+	{{Displayed math\|\|<math>4x = \pm 3x + \frac{\pi}{2} + 2n\pi\,\textrm{.}</math>}}

	This means that the solutions to the equation are		This means that the solutions to the equation are

		+	{{Displayed math\|\|<math>\left\{\begin{align}
		+	x &= \frac{\pi}{2} + 2n\pi\,,\\[5pt]
		+	x &= \frac{\pi}{14} + \frac{2}{7}\pi n\,,
		+	\end{align}\right.</math>}}

-	~~<math>\left\{ \begin{array}{*{35}l}~~	+	where ''n'' is an arbitrary integer.
-	~~x=\frac{\pi }{2}+2n\pi \\~~	+
-	~~x=\frac{\pi }{14}+\frac{2}{7}\pi~~ n \\	+
-	~~\end{array} \right.</math>~~	+
-	(	+
-	~~<math>n</math>~~	+
-	an arbitrary integer)	+

Ian um 13:20, 1. Okt. 2008

2008-10-01T13:20:21Z

← Nächstältere Version		Version vom 13:20, 1. Okt. 2008
Zeile 1:		Zeile 1:
-	{{~~NAVCONTENT_START}~~}	+	If we want to solve the equation
-	<~~center~~> ~~[[Image:4_4_7c-1(3).gif]]~~ </~~center~~>	+	<math>\text{cos 3}x=\text{sin 4}x</math>, we need an additional result which tells us for which values of
-	{~~{NAVCONTENT_STOP~~}}	+	<math>u</math>
-	{{~~NAVCONTENT_START}~~}	+	and
-	<~~center~~> ~~[[Image:4_4_7c-2(3)~~.~~gif]]~~ </~~center~~>	+	<math>v</math>
-	{~~{NAVCONTENT_STOP}~~}	+	the equality
-	{~~{NAVCONTENT_START}~~}	+	<math>\text{cos }u=\text{sin }v</math>
-	<~~center~~> ~~[[Image:4_4_7c-3(3)~~.~~gif]]~~ </~~center~~>	+	holds, but to get that we have to start with the equality
-	~~{{NAVCONTENT_STOP}}~~	+	<math>\cos u=\cos v</math>.
		+
		+	So, we start by looking at the equality
		+
		+
		+	<math>\cos u=\cos v</math>
		+
		+
		+	We know that for fixed
		+	<math>u</math>
		+	there are two angles
		+	<math>v=u\text{ }</math>
		+	and
		+	<math>v=-\text{u}</math>
		+	in the unit circle which have the cosine value
		+	<math>\cos u</math>, i.e. their
		+	<math>x</math>
		+	-coordinate is equal to
		+	<math>\cos u</math>.
		+

	[[Image:4_4_7_c1.gif\|center]]		[[Image:4_4_7_c1.gif\|center]]
		+
		+	Imagine now that the whole unit circle is rotated anti-clockwise an angle
		+	<math>{\pi }/{2}\;</math>. The line
		+	<math>x=\cos u</math>
		+	will become the line
		+	<math>y=\cos u</math>
		+	and the angles
		+	<math>u</math>
		+	and
		+	<math>-u</math>
		+	are rotated to
		+	<math>u+{\pi }/{2}\;</math>
		+	and
		+	<math>-u+{\pi }/{2}\;</math>, respectively.
		+

	[[Image:4_4_7_c2.gif\|center]]		[[Image:4_4_7_c2.gif\|center]]
		+
		+	The angles
		+	<math>u+{\pi }/{2}\;</math>
		+	and
		+	<math>-u+{\pi }/{2}\;</math>
		+	therefore have their
		+	<math>y</math>
		+	-coordinate, and hence sine value, equal to
		+	<math>\cos u</math>. In other words, the equality
		+
		+
		+	<math>\text{cos }u=\text{sin }v</math>
		+
		+
		+	holds for fixed
		+	<math>u</math>
		+	in the unit circle when
		+	<math>v=\pm u+{\pi }/{2}\;</math>, and more generally when
		+
		+
		+	<math>v=\pm u+\frac{\pi }{2}+2n\pi </math>
		+	(
		+	<math>n</math>
		+	an arbitrary integer).
		+
		+	For our equation
		+	<math>\text{cos 3}x=\text{sin 4}x</math>, this result means that
		+	<math>x\text{ }</math>
		+	must satisfy
		+
		+
		+	<math>4x=\pm 3x+\frac{\pi }{2}+2n\pi </math>
		+
		+
		+	This means that the solutions to the equation are
		+
		+
		+	<math>\left\{ \begin{array}{*{35}l}
		+	x=\frac{\pi }{2}+2n\pi \\
		+	x=\frac{\pi }{14}+\frac{2}{7}\pi n \\
		+	\end{array} \right.</math>
		+	(
		+	<math>n</math>
		+	an arbitrary integer)

Tekbot: Lösning 4.4:7c moved to Solution 4.4:7c: Robot: moved page

2008-09-09T10:08:23Z

Lösning 4.4:7c moved to Solution 4.4:7c: Robot: moved page

← Nächstältere Version

Version vom 10:08, 9. Sep. 2008

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

2008-08-21T07:21:19Z

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

← Nächstältere Version		Version vom 07:21, 21. Aug. 2008
Zeile 1:		Zeile 1:
	{{NAVCONTENT_START}}		{{NAVCONTENT_START}}
-	<center> [[~~Bild~~:4_4_7c-1(3).gif]] </center>	+	<center> [[Image:4_4_7c-1(3).gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}
	{{NAVCONTENT_START}}		{{NAVCONTENT_START}}
-	<center> [[~~Bild~~:4_4_7c-2(3).gif]] </center>	+	<center> [[Image:4_4_7c-2(3).gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}
	{{NAVCONTENT_START}}		{{NAVCONTENT_START}}
-	<center> [[~~Bild~~:4_4_7c-3(3).gif]] </center>	+	<center> [[Image:4_4_7c-3(3).gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}

-	[[~~Bild~~:4_4_7_c1.gif\|center]]	+	[[Image:4_4_7_c1.gif\|center]]

-	[[~~Bild~~:4_4_7_c2.gif\|center]]	+	[[Image:4_4_7_c2.gif\|center]]

Madde um 13:24, 19. Aug. 2008

2008-08-19T13:24:14Z

← Nächstältere Version		Version vom 13:24, 19. Aug. 2008
Zeile 8:		Zeile 8:
	<center> [[Bild:4_4_7c-3(3).gif]] </center>		<center> [[Bild:4_4_7c-3(3).gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}
		+
		+	[[Bild:4_4_7_c1.gif\|center]]
		+
		+	[[Bild:4_4_7_c2.gif\|center]]

← Nächstältere Version		Version vom 15:01, 19. Jun. 2009
Zeile 3:		Zeile 3:
	{{Abgesetzte Formel\|\|<math>\cos u=\cos v\,\textrm{.}</math>}}		{{Abgesetzte Formel\|\|<math>\cos u=\cos v\,\textrm{.}</math>}}

-	Wir wissen dass es für ein fixes ''u'' zwei ~~Winkeln im~~ Einheitskreis gibt für welche die Gleichung erfüllt ist, nämlich <math>v=u</math> und <math>v=-u</math>.	+	Wir wissen, dass es für ein fixes ''u'' zwei Winkel auf dem Einheitskreis gibt, für welche die Gleichung erfüllt ist, nämlich <math>v=u</math> und <math>v=-u</math>.

	[[Image:4_4_7_c1.gif\|center]]		[[Image:4_4_7_c1.gif\|center]]

-	Jetzt drehen wir ~~den~~ Einheitskreis den Winkel <math>\pi/2</math> gegen den ~~Uhrzeiger~~. Die Gerade <math>x=\cos u</math> ~~bekommt~~ dann <math>y=\cos u</math> und die ~~Winkeln~~ ''u'' und -''u'' ~~bekommen~~ <math>u+\pi/2</math> und <math>-u+\pi/2</math>,	+	Jetzt drehen wir auf dem Einheitskreis den Winkel <math>\pi/2</math> gegen den Uhrzeigersinn. Die Gerade <math>x=\cos u</math> wird dann <math>y=\cos u</math> und die Winkel ''u'' und -''u'' werden <math>u+\pi/2</math> und <math>-u+\pi/2</math>.

	[[Image:4_4_7_c2.gif\|center]]		[[Image:4_4_7_c2.gif\|center]]

-	Die ~~Winkeln~~ <math>u+\pi/2</math> und <math>-u+\pi/2</math> haben daher dieselbe ''y''-Koordinate, und daher denselben Sinus. Also ist die Gleichung	+	Die Winkel <math>u+\pi/2</math> und <math>-u+\pi/2</math> haben daher dieselbe ''y''-Koordinate und daher denselben Sinus. Also ist die Gleichung

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

-	für ein fixes ''u'' erfüllt wenn <math>v = \pm u + \pi/2</math>, und allgemein wenn	+	für ein fixes ''u'' erfüllt, wenn <math>v = \pm u + \pi/2</math>, und allgemein, wenn

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

-	In unseren Fall ist die Gleichung <math>\cos 3x = \sin 4x</math> erfüllt wenn	+	In unseren Fall ist die Gleichung <math>\cos 3x = \sin 4x</math> erfüllt, wenn

	{{Abgesetzte Formel\|\|<math>4x = \pm 3x + \frac{\pi}{2} + 2n\pi\,\textrm{.}</math>}}		{{Abgesetzte Formel\|\|<math>4x = \pm 3x + \frac{\pi}{2} + 2n\pi\,\textrm{.}</math>}}

-	~~und also bekommen~~ wir die allgemeinen Lösungen	+	Also erhalten wir die allgemeinen Lösungen

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

← Nächstältere Version		Version vom 09:01, 22. Okt. 2008
Zeile 4:		Zeile 4:
	So, we start by looking at the equality		So, we start by looking at the equality

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

	We know that for fixed ''u'' there are two angles <math>v=u</math> and <math>v=-u</math> in the unit circle which have the cosine value <math>\cos u</math>, i.e. their ''x''-coordinate is equal to <math>\cos u\,</math>.		We know that for fixed ''u'' there are two angles <math>v=u</math> and <math>v=-u</math> in the unit circle which have the cosine value <math>\cos u</math>, i.e. their ''x''-coordinate is equal to <math>\cos u\,</math>.
Zeile 16:		Zeile 16:
	The angles <math>u+\pi/2</math> and <math>-u+\pi/2</math> therefore have their ''y''-coordinate, and hence sine value, equal to <math>\cos u</math>. In other words, the equality		The angles <math>u+\pi/2</math> and <math>-u+\pi/2</math> therefore have their ''y''-coordinate, and hence sine value, equal to <math>\cos u</math>. In other words, the equality

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

	holds for fixed ''u'' in the unit circle when <math>v = \pm u + \pi/2</math>, and more generally when		holds for fixed ''u'' in the unit circle when <math>v = \pm u + \pi/2</math>, and more generally when

-	{{~~Displayed math~~\|\|<math>v = \pm u + \frac{\pi}{2} + 2n\pi\qquad</math>(''n'' is an arbitrary integer).}}	+	{{Abgesetzte Formel\|\|<math>v = \pm u + \frac{\pi}{2} + 2n\pi\qquad</math>(''n'' is an arbitrary integer).}}

	For our equation <math>\cos 3x = \sin 4x</math>, this result means that ''x'' must satisfy		For our equation <math>\cos 3x = \sin 4x</math>, this result means that ''x'' must satisfy

-	{{~~Displayed math~~\|\|<math>4x = \pm 3x + \frac{\pi}{2} + 2n\pi\,\textrm{.}</math>}}	+	{{Abgesetzte Formel\|\|<math>4x = \pm 3x + \frac{\pi}{2} + 2n\pi\,\textrm{.}</math>}}

	This means that the solutions to the equation are		This means that the solutions to the equation are

-	{{~~Displayed math~~\|\|<math>\left\{\begin{align}	+	{{Abgesetzte Formel\|\|<math>\left\{\begin{align}
	x &= \frac{\pi}{2} + 2n\pi\,,\\[5pt]		x &= \frac{\pi}{2} + 2n\pi\,,\\[5pt]
	x &= \frac{\pi}{14} + \frac{2}{7}\pi n\,,		x &= \frac{\pi}{14} + \frac{2}{7}\pi n\,,