Lösung 4.2:5b - Versionsgeschichte

Tek: Replaced figures with metapost figures

2009-08-24T14:00:46Z

Replaced figures with metapost figures

← Nächstältere Version		Version vom 14:00, 24. Aug. 2009
Zeile 1:		Zeile 1:
	Wir zeichnen den Winkel <math>225^{\circ} = 180^{\circ} + 45^{\circ}</math> am Einheitskreis und sehen, dass er den Winkel <math>45^{\circ}</math> zur negativen ''x''-Achse bildet.		Wir zeichnen den Winkel <math>225^{\circ} = 180^{\circ} + 45^{\circ}</math> am Einheitskreis und sehen, dass er den Winkel <math>45^{\circ}</math> zur negativen ''x''-Achse bildet.

-	~~[[Image~~:~~4_2_5_b1~~.~~gif\|~~center]]	+	<center>{{:4.2.5b - Solution - The unit circle with angle 180° + 45°}}</center>

	Also ist <math>\tan 225^{\circ} = \tan 45^{\circ}</math>, nachdem die beiden Geraden mit diesen Winkeln dieselbe Steigung haben:		Also ist <math>\tan 225^{\circ} = \tan 45^{\circ}</math>, nachdem die beiden Geraden mit diesen Winkeln dieselbe Steigung haben:
Zeile 7:		Zeile 7:
	{{Abgesetzte Formel\|\|<math>\tan 225^{\circ} = \tan 45^{\circ} = \frac{\sin 45^{\circ}}{\cos 45^{\circ}} = \frac{\dfrac{1}{\sqrt{2}}}{\dfrac{1}{\sqrt{2}}} = 1\,\textrm{.}</math>}}		{{Abgesetzte Formel\|\|<math>\tan 225^{\circ} = \tan 45^{\circ} = \frac{\sin 45^{\circ}}{\cos 45^{\circ}} = \frac{\dfrac{1}{\sqrt{2}}}{\dfrac{1}{\sqrt{2}}} = 1\,\textrm{.}</math>}}

-	~~[[Image~~:~~4_2_5_b2~~.~~gif\|~~center]]	+	<center>{{:4.2.5b - Solution - The unit circle with angles 225° and 45°}}</center>

Markus.bez um 13:25, 18. Jun. 2009

2009-06-18T13:25:09Z

← Nächstältere Version		Version vom 13:25, 18. Jun. 2009
Zeile 1:		Zeile 1:
-	Wir zeichnen den Winkel <math>225^{\circ} = 180^{\circ} + 45^{\circ}</math> am ~~einheitskreis,~~ und sehen dass er den Winkel <math>45^{\circ}</math> ~~mit der~~ negativen ''x''-Achse bildet.	+	Wir zeichnen den Winkel <math>225^{\circ} = 180^{\circ} + 45^{\circ}</math> am Einheitskreis und sehen, dass er den Winkel <math>45^{\circ}</math> zur negativen ''x''-Achse bildet.

	[[Image:4_2_5_b1.gif\|center]]		[[Image:4_2_5_b1.gif\|center]]

-	Also ist <math>\tan 225^{\circ} = \tan 45^{\circ}</math>, nachdem die beiden Geraden mit diesen Winkeln dieselbe Steigung haben,	+	Also ist <math>\tan 225^{\circ} = \tan 45^{\circ}</math>, nachdem die beiden Geraden mit diesen Winkeln dieselbe Steigung haben:

	{{Abgesetzte Formel\|\|<math>\tan 225^{\circ} = \tan 45^{\circ} = \frac{\sin 45^{\circ}}{\cos 45^{\circ}} = \frac{\dfrac{1}{\sqrt{2}}}{\dfrac{1}{\sqrt{2}}} = 1\,\textrm{.}</math>}}		{{Abgesetzte Formel\|\|<math>\tan 225^{\circ} = \tan 45^{\circ} = \frac{\sin 45^{\circ}}{\cos 45^{\circ}} = \frac{\dfrac{1}{\sqrt{2}}}{\dfrac{1}{\sqrt{2}}} = 1\,\textrm{.}</math>}}

	[[Image:4_2_5_b2.gif\|center]]		[[Image:4_2_5_b2.gif\|center]]

Martin um 16:01, 4. Apr. 2009

2009-04-04T16:01:54Z

← Nächstältere Version		Version vom 16:01, 4. Apr. 2009
Zeile 1:		Zeile 1:
-	~~If we draw the angle~~ <math>225^{\circ} = 180^{\circ} + 45^{\circ}</math> ~~on a unit circle~~, ~~we see that it makes an angle of~~ <math>45^{\circ}</math> ~~with the negative~~ ''x''-~~axis~~.	+	Wir zeichnen den Winkel <math>225^{\circ} = 180^{\circ} + 45^{\circ}</math> am einheitskreis, und sehen dass er den Winkel <math>45^{\circ}</math> mit der negativen ''x''-Achse bildet.

	[[Image:4_2_5_b1.gif\|center]]		[[Image:4_2_5_b1.gif\|center]]

-	~~This means that~~ <math>\tan 225^{\circ}~~</math>, which is the slope of the line that makes an angle of <math>45^{\circ}</math> with the positive ''x''-axis, equals <math>~~\tan 45^{\circ}</math>, ~~because the line which makes an angle of <math>45^{\circ}</math> has the same slope~~,	+	Also ist <math>\tan 225^{\circ} = \tan 45^{\circ}</math>, nachdem die beiden Geraden mit diesen Winkeln dieselbe Steigung haben,

	{{Abgesetzte Formel\|\|<math>\tan 225^{\circ} = \tan 45^{\circ} = \frac{\sin 45^{\circ}}{\cos 45^{\circ}} = \frac{\dfrac{1}{\sqrt{2}}}{\dfrac{1}{\sqrt{2}}} = 1\,\textrm{.}</math>}}		{{Abgesetzte Formel\|\|<math>\tan 225^{\circ} = \tan 45^{\circ} = \frac{\sin 45^{\circ}}{\cos 45^{\circ}} = \frac{\dfrac{1}{\sqrt{2}}}{\dfrac{1}{\sqrt{2}}} = 1\,\textrm{.}</math>}}

	[[Image:4_2_5_b2.gif\|center]]		[[Image:4_2_5_b2.gif\|center]]

Tekbot: hat „Solution 4.2:5b“ nach „Lösung 4.2:5b“ verschoben: Robot: moved page

2008-10-22T14:58:24Z

hat „Solution 4.2:5b“ nach „Lösung 4.2:5b“ verschoben: Robot: moved page

← Nächstältere Version

Version vom 14:58, 22. Okt. 2008

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

2008-10-22T08:52:58Z

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

← Nächstältere Version		Version vom 08:52, 22. Okt. 2008
Zeile 5:		Zeile 5:
	This means that <math>\tan 225^{\circ}</math>, which is the slope of the line that makes an angle of <math>45^{\circ}</math> with the positive ''x''-axis, equals <math>\tan 45^{\circ}</math>, because the line which makes an angle of <math>45^{\circ}</math> has the same slope,		This means that <math>\tan 225^{\circ}</math>, which is the slope of the line that makes an angle of <math>45^{\circ}</math> with the positive ''x''-axis, equals <math>\tan 45^{\circ}</math>, because the line which makes an angle of <math>45^{\circ}</math> has the same slope,

-	{{~~Displayed math~~\|\|<math>\tan 225^{\circ} = \tan 45^{\circ} = \frac{\sin 45^{\circ}}{\cos 45^{\circ}} = \frac{\dfrac{1}{\sqrt{2}}}{\dfrac{1}{\sqrt{2}}} = 1\,\textrm{.}</math>}}	+	{{Abgesetzte Formel\|\|<math>\tan 225^{\circ} = \tan 45^{\circ} = \frac{\sin 45^{\circ}}{\cos 45^{\circ}} = \frac{\dfrac{1}{\sqrt{2}}}{\dfrac{1}{\sqrt{2}}} = 1\,\textrm{.}</math>}}

	[[Image:4_2_5_b2.gif\|center]]		[[Image:4_2_5_b2.gif\|center]]

Tek um 11:08, 9. Okt. 2008

2008-10-09T11:08:20Z

← Nächstältere Version		Version vom 11:08, 9. Okt. 2008
Zeile 1:		Zeile 1:
-	If we draw the angle	+	If we draw the angle <math>225^{\circ} = 180^{\circ} + 45^{\circ}</math> on a unit circle, we see that it makes an angle of <math>45^{\circ}</math> with the negative ''x''-axis.
-	<math>~~\text{~~225}^{\circ ~~}\text{~~ }=~~\text{ 18}0~~^{\circ ~~}\text{~~ }+~~\text{~~ 45}^{\circ }</math>	+
-	on a unit circle, we see that it makes an angle of	+
-	<math>~~\text{~~45}^{\circ }</math>	+
-	with the negative	+
-	~~<math>~~x~~</math>~~	+
-	-axis.	+

	[[Image:4_2_5_b1.gif\|center]]		[[Image:4_2_5_b1.gif\|center]]

-	This means that	+	This means that <math>\tan 225^{\circ}</math>, which is the slope of the line that makes an angle of <math>45^{\circ}</math> with the positive ''x''-axis, equals <math>\tan 45^{\circ}</math>, because the line which makes an angle of <math>45^{\circ}</math> has the same slope,
-	<math>\~~text{~~tan 225}^{\circ }</math>, which is the ~~gradient~~ of the line that makes an angle of	+
-	<math>~~\text{~~45}^{\circ }</math>	+
-	with the positive	+
-	~~<math>~~x~~</math>~~	+
-	-axis, equals	+
-	<math>\~~text{~~tan ~~225}~~^{\circ }</math>, because the line which makes an angle of	+
-	<math>~~\text{~~45}^{\circ }</math>	+
-	has the same slope:	+
-		+
-		+
-	~~<math>\tan 225^{\circ }\text{ }=\tan \text{45}^{\circ }=\frac{\sin \text{45}^{\circ }}{\cos \text{45}^{\circ }}=\frac{\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}}=1</math>~~	+
-		+

		+	{{Displayed math\|\|<math>\tan 225^{\circ} = \tan 45^{\circ} = \frac{\sin 45^{\circ}}{\cos 45^{\circ}} = \frac{\dfrac{1}{\sqrt{2}}}{\dfrac{1}{\sqrt{2}}} = 1\,\textrm{.}</math>}}

	[[Image:4_2_5_b2.gif\|center]]		[[Image:4_2_5_b2.gif\|center]]

Ian um 08:04, 29. Sep. 2008

2008-09-29T08:04:44Z

← Nächstältere Version		Version vom 08:04, 29. Sep. 2008
Zeile 1:		Zeile 1:
-	{{~~NAVCONTENT_START~~}}	+	If we draw the angle
-	~~<center> [[Image:4_2_5b.gif]]~~ </~~center~~>	+	<math>\text{225}^{\circ }\text{ }=\text{ 18}0^{\circ }\text{ }+\text{ 45}^{\circ }</math>
-	{{~~NAVCONTENT_STOP}~~}	+	on a unit circle, we see that it makes an angle of
		+	<math>\text{45}^{\circ }</math>
		+	with the negative
		+	<math>x</math>
		+	-axis.
		+
	[[Image:4_2_5_b1.gif\|center]]		[[Image:4_2_5_b1.gif\|center]]
		+
		+	This means that
		+	<math>\text{tan 225}^{\circ }</math>, which is the gradient of the line that makes an angle of
		+	<math>\text{45}^{\circ }</math>
		+	with the positive
		+	<math>x</math>
		+	-axis, equals
		+	<math>\text{tan 225}^{\circ }</math>, because the line which makes an angle of
		+	<math>\text{45}^{\circ }</math>
		+	has the same slope:
		+
		+
		+	<math>\tan 225^{\circ }\text{ }=\tan \text{45}^{\circ }=\frac{\sin \text{45}^{\circ }}{\cos \text{45}^{\circ }}=\frac{\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}}=1</math>
		+
		+
		+
	[[Image:4_2_5_b2.gif\|center]]		[[Image:4_2_5_b2.gif\|center]]

Tekbot: Lösning 4.2:5b moved to Solution 4.2:5b: Robot: moved page

2008-09-09T09:48:03Z

Lösning 4.2:5b moved to Solution 4.2:5b: Robot: moved page

← Nächstältere Version

Version vom 09:48, 9. Sep. 2008

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

2008-08-21T07:11:09Z

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

← Nächstältere Version		Version vom 07:11, 21. Aug. 2008
Zeile 1:		Zeile 1:
	{{NAVCONTENT_START}}		{{NAVCONTENT_START}}
-	<center> [[~~Bild~~:4_2_5b.gif]] </center>	+	<center> [[Image:4_2_5b.gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}
-	[[~~Bild~~:4_2_5_b1.gif\|center]]	+	[[Image:4_2_5_b1.gif\|center]]
-	[[~~Bild~~:4_2_5_b2.gif\|center]]	+	[[Image:4_2_5_b2.gif\|center]]

Madde um 15:34, 15. Aug. 2008

2008-08-15T15:34:12Z

← Nächstältere Version		Version vom 15:34, 15. Aug. 2008
Zeile 2:		Zeile 2:
	<center> [[Bild:4_2_5b.gif]] </center>		<center> [[Bild:4_2_5b.gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}
		+	[[Bild:4_2_5_b1.gif\|center]]
		+	[[Bild:4_2_5_b2.gif\|center]]