Lösung 4.4:5b - Versionsgeschichte

Tek: Replaced figure with metapost figure

2009-08-25T13:26:51Z

Replaced figure with metapost figure

← Nächstältere Version		Version vom 13:26, 25. Aug. 2009
Zeile 7:		Zeile 7:
	{{Abgesetzte Formel\|\|<math>v=u\qquad\text{und}\qquad v=u+\pi\,\textrm{.}</math>}}		{{Abgesetzte Formel\|\|<math>v=u\qquad\text{und}\qquad v=u+\pi\,\textrm{.}</math>}}

-	~~[[Image~~:~~4_4_5_b_de~~.~~gif\|~~center]]	+	<center>{{:4.4.5b - Solution - Two unit circles with angles v = u and v = π - u, respectively}}</center>

	Die allgemeine Lösung ist		Die allgemeine Lösung ist

Bayerer um 15:52, 30. Jul. 2009

2009-07-30T15:52:45Z

← Nächstältere Version		Version vom 15:52, 30. Jul. 2009
Zeile 7:		Zeile 7:
	{{Abgesetzte Formel\|\|<math>v=u\qquad\text{und}\qquad v=u+\pi\,\textrm{.}</math>}}		{{Abgesetzte Formel\|\|<math>v=u\qquad\text{und}\qquad v=u+\pi\,\textrm{.}</math>}}

-	[[Image:~~4_4_5_b~~.gif\|center]]	+	[[Image:4_4_5_b_de.gif\|center]]

	Die allgemeine Lösung ist		Die allgemeine Lösung ist

Markus.bez um 14:34, 19. Jun. 2009

2009-06-19T14:34:19Z

← Nächstältere Version		Version vom 14:34, 19. Jun. 2009
Zeile 3:		Zeile 3:
	{{Abgesetzte Formel\|\|<math>\tan u=\tan v</math>}}		{{Abgesetzte Formel\|\|<math>\tan u=\tan v</math>}}

-	Diese ~~Gleichungist~~ erfüllt wenn	+	Diese Gleichung ist erfüllt, wenn

	{{Abgesetzte Formel\|\|<math>v=u\qquad\text{und}\qquad v=u+\pi\,\textrm{.}</math>}}		{{Abgesetzte Formel\|\|<math>v=u\qquad\text{und}\qquad v=u+\pi\,\textrm{.}</math>}}
Zeile 19:		Zeile 19:
	erhalten wir die Lösungen		erhalten wir die Lösungen

-	{{Abgesetzte Formel\|\|<math>4x = x+n\pi </math>}}	+	{{Abgesetzte Formel\|\|<math>4x = x+n\pi\,. </math>}}

-	Lösen wir diese Gleichung für ''x'' erhalten wir	+	Lösen wir diese Gleichung für ''x'', erhalten wir

	{{Abgesetzte Formel\|\|<math>x = \tfrac{1}{3}n\pi</math>}}		{{Abgesetzte Formel\|\|<math>x = \tfrac{1}{3}n\pi</math>}}

Martin um 11:18, 7. Apr. 2009

2009-04-07T11:18:15Z

← Nächstältere Version		Version vom 11:18, 7. Apr. 2009
Zeile 1:		Zeile 1:
-	~~Let's first investigate when the equality~~	+	Wir betrachten zuerst die Gleichung

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

-	is satisfied. Because <math>\tan u</math> can be interpreted as the slope (gradient) of the line which makes an angle ''u'' with the positive ''x''-axis, we see that for a fixed value of <math>\tan u</math>, there are two angles ''v'' in the unit circle with this slope,	+	Diese Gleichungist erfüllt wenn

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

	[[Image:4_4_5_b.gif\|center]]		[[Image:4_4_5_b.gif\|center]]

-	~~The angle ''v'' has the same slope after every half turn, so if we add multiples of~~	+	Die allgemeine Lösung ist
-	~~<math>\pi</math> to ''u'', we will obtain all the angles ''v'' which satisfy the equality~~	+

	{{Abgesetzte Formel\|\|<math>v=u+n\pi\,,</math>}}		{{Abgesetzte Formel\|\|<math>v=u+n\pi\,,</math>}}

-	~~where ''n'' is an arbitrary integer.~~	+	Für unsere Gleichung
-		+
-	~~If we apply this result to the equation~~	+

	{{Abgesetzte Formel\|\|<math>\tan x=\tan 4x</math>}}		{{Abgesetzte Formel\|\|<math>\tan x=\tan 4x</math>}}

-	~~we see that the solutions are given by~~	+	erhalten wir die Lösungen

-	{{Abgesetzte Formel\|\|<math>4x = x+n\pi~~\qquad\text{(n is an arbitrary integer),}~~</math>}}	+	{{Abgesetzte Formel\|\|<math>4x = x+n\pi </math>}}

-	~~and solving for~~ ''x'' ~~gives~~	+	Lösen wir diese Gleichung für ''x'' erhalten wir

-	{{Abgesetzte Formel\|\|<math>x = \tfrac{1}{3}n\pi~~\qquad\text{(n is an arbitrary integer).}~~</math>}}	+	{{Abgesetzte Formel\|\|<math>x = \tfrac{1}{3}n\pi</math>}}

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

2008-10-22T15:16:24Z

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

← Nächstältere Version

Version vom 15:16, 22. Okt. 2008

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

2008-10-22T08:59:58Z

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

← Nächstältere Version		Version vom 08:59, 22. Okt. 2008
Zeile 1:		Zeile 1:
	Let's first investigate when the equality		Let's first investigate when the equality

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

	is satisfied. Because <math>\tan u</math> can be interpreted as the slope (gradient) of the line which makes an angle ''u'' with the positive ''x''-axis, we see that for a fixed value of <math>\tan u</math>, there are two angles ''v'' in the unit circle with this slope,		is satisfied. Because <math>\tan u</math> can be interpreted as the slope (gradient) of the line which makes an angle ''u'' with the positive ''x''-axis, we see that for a fixed value of <math>\tan u</math>, there are two angles ''v'' in the unit circle with this slope,

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

	[[Image:4_4_5_b.gif\|center]]		[[Image:4_4_5_b.gif\|center]]
Zeile 12:		Zeile 12:
	<math>\pi</math> to ''u'', we will obtain all the angles ''v'' which satisfy the equality		<math>\pi</math> to ''u'', we will obtain all the angles ''v'' which satisfy the equality

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

	where ''n'' is an arbitrary integer.		where ''n'' is an arbitrary integer.
Zeile 18:		Zeile 18:
	If we apply this result to the equation		If we apply this result to the equation

-	{{~~Displayed math~~\|\|<math>\tan x=\tan 4x</math>}}	+	{{Abgesetzte Formel\|\|<math>\tan x=\tan 4x</math>}}

	we see that the solutions are given by		we see that the solutions are given by

-	{{~~Displayed math~~\|\|<math>4x = x+n\pi\qquad\text{(n is an arbitrary integer),}</math>}}	+	{{Abgesetzte Formel\|\|<math>4x = x+n\pi\qquad\text{(n is an arbitrary integer),}</math>}}

	and solving for ''x'' gives		and solving for ''x'' gives

-	{{~~Displayed math~~\|\|<math>x = \tfrac{1}{3}n\pi\qquad\text{(n is an arbitrary integer).}</math>}}	+	{{Abgesetzte Formel\|\|<math>x = \tfrac{1}{3}n\pi\qquad\text{(n is an arbitrary integer).}</math>}}

Tek um 14:02, 13. Okt. 2008

2008-10-13T14:02:44Z

← Nächstältere Version		Version vom 14:02, 13. Okt. 2008
Zeile 1:		Zeile 1:
	Let's first investigate when the equality		Let's first investigate when the equality

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

-	~~<math>\tan u=\tan v</math>~~	+	is satisfied. Because <math>\tan u</math> can be interpreted as the slope (gradient) of the line which makes an angle ''u'' with the positive ''x''-axis, we see that for a fixed value of <math>\tan u</math>, there are two angles ''v'' in the unit circle with this slope,
-		+
-		+
-	is satisfied. Because	+
-	<math>u</math>	+
-	can be interpreted as the slope (gradient) of the line which makes an angle	+
-	~~<math>~~u~~</math>~~	+
-	with the positive	+
-	~~<math>~~x~~</math>~~	+
-	-axis, we see that for a fixed value of tan u, there are two angles	+
-	~~<math>~~v~~</math>~~	+
-	in the unit circle with this slope:	+
-		+
-		+
-	~~<math>v=u</math>~~	+
-	~~and~~	+
-	~~<math>v=u+\pi </math>~~	+

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

	[[Image:4_4_5_b.gif\|center]]		[[Image:4_4_5_b.gif\|center]]

-	slope	+	The angle ''v'' has the same slope after every half turn, so if we add multiples of
-	<math>=\~~text{ tan }u~~</math>	+	<math>\pi</math> to ''u'', we will obtain all the angles ''v'' which satisfy the equality
-	~~slope~~	+
-	~~<math>=\text{ tan }~~u~~</math>~~	+

		+	{{Displayed math\|\|<math>v=u+n\pi\,,</math>}}

-	~~The angle~~	+	where ''n'' is an arbitrary integer.
-	~~<math>v</math>~~	+
-	~~has the same slope after every half turn, so if we add multiples of~~	+
-	~~<math>\pi \text{ }</math>~~	+
-	to	+
-	~~<math>u</math>, we will obtain all the angles~~	+
-	~~<math>v</math>~~	+
-	~~which satisfy the equality~~	+
-		+
-		+
-	~~<math>v=u+n\pi </math>~~	+
-		+
-		+
-	where	+
-	~~<math>~~n~~</math>~~	+
-	is an arbitrary integer.	+

	If we apply this result to the equation		If we apply this result to the equation

-		+	{{Displayed math\|\|<math>\tan x=\tan 4x</math>}}
-	<math>\tan x=\tan 4x</math>	+
-		+

	we see that the solutions are given by		we see that the solutions are given by

		+	{{Displayed math\|\|<math>4x = x+n\pi\qquad\text{(n is an arbitrary integer),}</math>}}

-	~~<math>4x=x+n\pi </math>~~	+	and solving for ''x'' gives
-	(	+
-	~~<math>n</math>~~	+
-	~~an arbitrary integer),~~	+
-		+
-	and solving for	+
-	~~<math>~~x~~</math>~~	+
-	gives	+
-		+

-	<math>x=\~~frac~~{1}{3}n\pi ~~</math>~~	+	{{Displayed math\|\|<math>x = \tfrac{1}{3}n\pi\qquad\text{(n is an arbitrary integer).}</math>}}
-	(	+
-	~~<math>~~n~~</math>~~	+
-	an arbitrary integer).	+

Ian um 11:06, 1. Okt. 2008

2008-10-01T11:06:30Z

← Nächstältere Version		Version vom 11:06, 1. Okt. 2008
Zeile 1:		Zeile 1:
-	~~{{NAVCONTENT_START}}~~	+	Let's first investigate when the equality
-	<~~center~~> ~~[[Image:4_4_5b-1~~(2)~~.gif]]~~ </~~center~~>	+
-	~~{{NAVCONTENT_STOP}}~~	+
-	~~{{NAVCONTENT_START}}~~	+	<math>\tan u=\tan v</math>
-	<~~center~~> ~~[[Image~~:~~4_4_5b-2(2).gif]]~~ </~~center~~>	+
-	~~{{NAVCONTENT_STOP}}~~	+
		+	is satisfied. Because
		+	<math>u</math>
		+	can be interpreted as the slope (gradient) of the line which makes an angle
		+	<math>u</math>
		+	with the positive
		+	<math>x</math>
		+	-axis, we see that for a fixed value of tan u, there are two angles
		+	<math>v</math>
		+	in the unit circle with this slope:
		+
		+
		+	<math>v=u</math>
		+	and
		+	<math>v=u+\pi </math>
		+

	[[Image:4_4_5_b.gif\|center]]		[[Image:4_4_5_b.gif\|center]]
		+
		+	slope
		+	<math>=\text{ tan }u</math>
		+	slope
		+	<math>=\text{ tan }u</math>
		+
		+
		+	The angle
		+	<math>v</math>
		+	has the same slope after every half turn, so if we add multiples of
		+	<math>\pi \text{ }</math>
		+	to
		+	<math>u</math>, we will obtain all the angles
		+	<math>v</math>
		+	which satisfy the equality
		+
		+
		+	<math>v=u+n\pi </math>
		+
		+
		+	where
		+	<math>n</math>
		+	is an arbitrary integer.
		+
		+	If we apply this result to the equation
		+
		+
		+	<math>\tan x=\tan 4x</math>
		+
		+
		+	we see that the solutions are given by
		+
		+
		+	<math>4x=x+n\pi </math>
		+	(
		+	<math>n</math>
		+	an arbitrary integer),
		+
		+	and solving for
		+	<math>x</math>
		+	gives
		+
		+
		+	<math>x=\frac{1}{3}n\pi </math>
		+	(
		+	<math>n</math>
		+	an arbitrary integer).

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

2008-09-09T10:06:03Z

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

← Nächstältere Version

Version vom 10:06, 9. Sep. 2008

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

2008-08-21T07:20:09Z

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_5b-1(2).gif]] </center>	+	<center> [[Image:4_4_5b-1(2).gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}
	{{NAVCONTENT_START}}		{{NAVCONTENT_START}}
-	<center> [[~~Bild~~:4_4_5b-2(2).gif]] </center>	+	<center> [[Image:4_4_5b-2(2).gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}

-	[[~~Bild~~:4_4_5_b.gif\|center]]	+	[[Image:4_4_5_b.gif\|center]]