Lösung 3.3:1c - Versionsgeschichte

Sekretariat1 um 07:48, 1. Sep. 2009

Sekretariat1 — Tue, 01 Sep 2009 07:48:04 GMT

← Nächstältere Version		Version vom 07:48, 1. Sep. 2009
Zeile 1:		Zeile 1:
-	Wir bringen zuerst <math>4\sqrt{3}-4i</math> ~~auf~~ Polarform und verwenden dann den Moivreschen Satz.	+	Wir bringen zuerst <math>4\sqrt{3}-4i</math> in Polarform und verwenden dann den Moivreschen Satz.

	<center>[[Image:3_3_1_c.gif]] [[Image:3_3_1_c_text.gif]]</center>		<center>[[Image:3_3_1_c.gif]] [[Image:3_3_1_c_text.gif]]</center>

Moni um 20:54, 22. Aug. 2009

Moni — Sat, 22 Aug 2009 20:54:33 GMT

← Nächstältere Version		Version vom 20:54, 22. Aug. 2009
Zeile 1:		Zeile 1:
-	Wir bringen zuerst <math>4\sqrt{3}-4i</math> auf Polarform, und verwenden dann den Moivreschen Satz.	+	Wir bringen zuerst <math>4\sqrt{3}-4i</math> auf Polarform und verwenden dann den Moivreschen Satz.

	<center>[[Image:3_3_1_c.gif]] [[Image:3_3_1_c_text.gif]]</center>		<center>[[Image:3_3_1_c.gif]] [[Image:3_3_1_c_text.gif]]</center>

Martin um 14:45, 18. Mai 2009

Martin — Mon, 18 May 2009 14:45:18 GMT

← Nächstältere Version		Version vom 14:45, 18. Mai 2009
Zeile 1:		Zeile 1:
-	~~The calculation follows a fairly set pattern. We write the number~~ <math>4\sqrt{3}-4i</math> ~~in polar form and then use de Moivre's formula~~.	+	Wir bringen zuerst <math>4\sqrt{3}-4i</math> auf Polarform, und verwenden dann den Moivreschen Satz.

	<center>[[Image:3_3_1_c.gif]] [[Image:3_3_1_c_text.gif]]</center>		<center>[[Image:3_3_1_c.gif]] [[Image:3_3_1_c_text.gif]]</center>

-	~~This gives~~	+	Dies ergibt

	{{Abgesetzte Formel\|\|<math>4\sqrt{3}-4i = 8\Bigl(\cos\Bigl(-\frac{\pi}{6}\Bigr) + i\sin\Bigl(-\frac{\pi}{6}\Bigr)\Bigr)</math>}}		{{Abgesetzte Formel\|\|<math>4\sqrt{3}-4i = 8\Bigl(\cos\Bigl(-\frac{\pi}{6}\Bigr) + i\sin\Bigl(-\frac{\pi}{6}\Bigr)\Bigr)</math>}}

-	~~and then we get, on using de Moivre's formula,~~	+	und durch den Moivreschen Satz erhalten wir

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

Tekbot: Solution 3.3:1c moved to Lösung 3.3:1c: Robot: moved page

Tekbot — Wed, 11 Mar 2009 10:43:15 GMT

Solution 3.3:1c moved to Lösung 3.3:1c: Robot: moved page

← Nächstältere Version

Version vom 10:43, 11. Mär. 2009

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

Tekbot — Tue, 10 Mar 2009 13:11:04 GMT

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

← Nächstältere Version		Version vom 13:11, 10. Mär. 2009
Zeile 5:		Zeile 5:
	This gives		This gives

-	{{~~Displayed math~~\|\|<math>4\sqrt{3}-4i = 8\Bigl(\cos\Bigl(-\frac{\pi}{6}\Bigr) + i\sin\Bigl(-\frac{\pi}{6}\Bigr)\Bigr)</math>}}	+	{{Abgesetzte Formel\|\|<math>4\sqrt{3}-4i = 8\Bigl(\cos\Bigl(-\frac{\pi}{6}\Bigr) + i\sin\Bigl(-\frac{\pi}{6}\Bigr)\Bigr)</math>}}

	and then we get, on using de Moivre's formula,		and then we get, on using de Moivre's formula,

-	{{~~Displayed math~~\|\|<math>\begin{align}	+	{{Abgesetzte Formel\|\|<math>\begin{align}
	\bigl(4\sqrt{3}-4i\bigr)^{22}		\bigl(4\sqrt{3}-4i\bigr)^{22}
	&= 8^{22}\Bigl(\cos\Bigl(22\cdot\Bigl(-\frac{\pi}{6}\Bigr)\Bigr) + i\sin\Bigl(22\cdot\Bigl(-\frac{\pi}{6}\Bigr)\Bigr)\Bigr)\\[5pt]		&= 8^{22}\Bigl(\cos\Bigl(22\cdot\Bigl(-\frac{\pi}{6}\Bigr)\Bigr) + i\sin\Bigl(22\cdot\Bigl(-\frac{\pi}{6}\Bigr)\Bigr)\Bigr)\\[5pt]

Tek um 09:06, 30. Okt. 2008

Tek — Thu, 30 Oct 2008 09:06:43 GMT

← Nächstältere Version		Version vom 09:06, 30. Okt. 2008
Zeile 1:		Zeile 1:
-	The calculation follows a fairly set pattern. We write the number	+	The calculation follows a fairly set pattern. We write the number <math>4\sqrt{3}-4i</math> in polar form and then use de Moivre's formula.
-	<math>4\sqrt{3}-4i</math>	+
-	in polar form and then use de Moivre's formula.	+
-		+
-		+
-	~~[[Image:3_3_1_c.gif]] [[Image:3_3_1_c_text.gif]]~~	+

		+	<center>[[Image:3_3_1_c.gif]] [[Image:3_3_1_c_text.gif]]</center>

	This gives		This gives

-		+	{{Displayed math\|\|<math>4\sqrt{3}-4i = 8\Bigl(\cos\Bigl(-\frac{\pi}{6}\Bigr) + i\sin\Bigl(-\frac{\pi}{6}\Bigr)\Bigr)</math>}}
-	<math>4\sqrt{3}-4i=8\~~left~~( \cos \~~left~~( -\frac{\pi }{6} \~~right~~)+i\sin \~~left~~( -\frac{\pi }{6} \~~right~~) \~~right~~)</math>	+
-		+

	and then we get, on using de Moivre's formula,		and then we get, on using de Moivre's formula,

-		+	{{Displayed math\|\|<math>\begin{align}
-	<math>\begin{align}	+	\bigl(4\sqrt{3}-4i\bigr)^{22}
-	& \~~left~~( 4\sqrt{3}-4i \~~right~~)^{22}=8^{22}\~~left~~( \cos \~~left~~( 22\~~centerdot~~ \~~left~~( -\frac{\pi }{6} \~~right~~) \~~right~~)+i\sin \~~left~~( 22\~~centerdot~~ \~~left~~( -\frac{\pi }{6} \~~right~~) \~~right~~) \~~right~~) \\	+	&= 8^{22}\Bigl(\cos\Bigl(22\cdot\Bigl(-\frac{\pi}{6}\Bigr)\Bigr) + i\sin\Bigl(22\cdot\Bigl(-\frac{\pi}{6}\Bigr)\Bigr)\Bigr)\\[5pt]
-	& =\~~left~~( 2^{3} \~~right~~)^{22}\~~left~~( \cos \~~left~~( -\frac{11\pi }{3} \~~right~~)+i\sin \~~left~~( -\frac{11\pi }{3} \~~right~~) \~~right~~) \\	+	&= \bigl(2^3\bigr)^{22}\Bigl(\cos\Bigl(-\frac{11\pi}{3}\Bigr) + i\sin\Bigl(-\frac{11\pi}{3}\Bigr)\Bigr)\\[5pt]
-	& =2^{3\~~centerdot~~ 22}\~~left~~( \cos \~~left~~( -\frac{12\pi -\pi }{3} \~~right~~)+i\sin \~~left~~( -\frac{12\pi -\pi }{3} \~~right~~) \~~right~~) \\	+	&= 2^{3\cdot 22}\Bigl(\cos\Bigl(-\frac{12\pi -\pi }{3}\Bigr) + i\sin\Bigl(-\frac{12\pi -\pi}{3}\Bigr)\Bigr)\\[5pt]
-	& =2^{66}\~~left~~( \cos \~~left~~( -4\pi +\frac{\pi }{3} \~~right~~)+i\sin \~~left~~( -4\pi +\frac{\pi }{3} \~~right~~) \~~right~~) \\	+	&= 2^{66}\Bigl(\cos\Bigl(-4\pi+\frac{\pi}{3}\Bigr) + i\sin\Bigl(-4\pi+\frac{\pi}{3} \Bigr)\Bigr)\\[5pt]
-	& =2^{66}\~~left~~( \cos ~~\left(~~ \frac{\pi }{3} ~~\right)~~+i\sin ~~\left(~~ \frac{\pi }{3} \~~right) \right~~) \\	+	&= 2^{66}\Bigl(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\Bigr)\\[5pt]
-	& =2^{66}\~~left~~( \frac{1}{2}+i\frac{\sqrt{3}}{2} \~~right~~)=2^{65}~~\left~~( 1+i\sqrt{3} \~~right~~) \\	+	&= 2^{66}\Bigl(\frac{1}{2} + i\frac{\sqrt{3}}{2}\Bigr)\\[5pt]
-	\end{align}</math>	+	&= 2^{65}(1+i\sqrt{3}\,)\,\textrm{.}
		+	\end{align}</math>}}

Ian um 07:28, 24. Okt. 2008

Ian — Fri, 24 Oct 2008 07:28:24 GMT

← Nächstältere Version		Version vom 07:28, 24. Okt. 2008
Zeile 1:		Zeile 1:
-	~~{{NAVCONTENT_START}}~~	+	The calculation follows a fairly set pattern. We write the number
-	<~~center~~> ~~[[Image:3_3_1c.gif]]~~ </~~center~~>	+	<math>4\sqrt{3}-4i</math>
-	~~{{NAVCONTENT_STOP}}~~	+	in polar form and then use de Moivre's formula.
		+

	[[Image:3_3_1_c.gif]] [[Image:3_3_1_c_text.gif]]		[[Image:3_3_1_c.gif]] [[Image:3_3_1_c_text.gif]]
		+
		+
		+	This gives
		+
		+
		+	<math>4\sqrt{3}-4i=8\left( \cos \left( -\frac{\pi }{6} \right)+i\sin \left( -\frac{\pi }{6} \right) \right)</math>
		+
		+
		+	and then we get, on using de Moivre's formula,
		+
		+
		+	<math>\begin{align}
		+	& \left( 4\sqrt{3}-4i \right)^{22}=8^{22}\left( \cos \left( 22\centerdot \left( -\frac{\pi }{6} \right) \right)+i\sin \left( 22\centerdot \left( -\frac{\pi }{6} \right) \right) \right) \\
		+	& =\left( 2^{3} \right)^{22}\left( \cos \left( -\frac{11\pi }{3} \right)+i\sin \left( -\frac{11\pi }{3} \right) \right) \\
		+	& =2^{3\centerdot 22}\left( \cos \left( -\frac{12\pi -\pi }{3} \right)+i\sin \left( -\frac{12\pi -\pi }{3} \right) \right) \\
		+	& =2^{66}\left( \cos \left( -4\pi +\frac{\pi }{3} \right)+i\sin \left( -4\pi +\frac{\pi }{3} \right) \right) \\
		+	& =2^{66}\left( \cos \left( \frac{\pi }{3} \right)+i\sin \left( \frac{\pi }{3} \right) \right) \\
		+	& =2^{66}\left( \frac{1}{2}+i\frac{\sqrt{3}}{2} \right)=2^{65}\left( 1+i\sqrt{3} \right) \\
		+	\end{align}</math>

Tekbot: Lösning 3.3:1c moved to Solution 3.3:1c: Robot: moved page

Tekbot — Tue, 16 Sep 2008 15:03:41 GMT

Lösning 3.3:1c moved to Solution 3.3:1c: Robot: moved page

← Nächstältere Version

Version vom 15:03, 16. Sep. 2008

Madde um 12:42, 21. Aug. 2008

Madde — Thu, 21 Aug 2008 12:42:56 GMT

← Nächstältere Version		Version vom 12:42, 21. Aug. 2008
Zeile 2:		Zeile 2:
	<center> [[Image:3_3_1c.gif]] </center>		<center> [[Image:3_3_1c.gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}
		+
		+	[[Image:3_3_1_c.gif]] [[Image:3_3_1_c_text.gif]]

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

Tekbot — Thu, 21 Aug 2008 08:11:34 GMT

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

← Nächstältere Version		Version vom 08:11, 21. Aug. 2008
Zeile 1:		Zeile 1:
	{{NAVCONTENT_START}}		{{NAVCONTENT_START}}
-	<center> [[~~Bild~~:3_3_1c.gif]] </center>	+	<center> [[Image:3_3_1c.gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}