Solution 4.4:2d - Revision history

Tek at 14:32, 10 October 2008

2008-10-10T14:32:03Z

←Older revision		Revision as of 14:32, 10 October 2008
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-	Apart from the fact that there is a	+	Apart from the fact that there is a <math>5x</math>, this is a normal trigonometric equation of the type <math>\sin y = a\,</math>. If we are only interested in solutions which satisfy <math>0\le 5x\le 2\pi</math>, then a sketch of the unit circle shows that there are two such solutions, <math>5x = \pi/4</math> and the reflectionally symmetric solution <math>5x = \pi - \pi/4 = 3\pi/4\,</math>.
-	<math>~~\text{5}x~~</math>, this is a normal trigonometric equation of the type	+
-	<math>\~~text{~~sin }y~~\text{ }~~=a</math> . If we are only interested in solutions which satisfy	+
-	<math>0\le ~~\text{5}x~~\le ~~\text{~~2}\pi </math>, then a sketch of the unit circle shows that there are two such solutions,	+
-	<math>~~\text{5}x~~=~~\text{ }\frac{~~\pi }{4}</math>	+
-	and the reflectionally symmetric solution	+
-	<math>~~\text{5}x~~=~~\text{ }~~\pi ~~\text{~~-~~}\frac{~~\pi }{4}=~~\frac{~~3\pi }{4}</math>.	+
-		+

	[[Image:4_4_2_d.gif\|center]]		[[Image:4_4_2_d.gif\|center]]

-	All of the equation's solutions are obtained from all values of 5x which differ by a multiple of ~~2π from either of these two solutions:~~	+	All of the equation's solutions are obtained from all values of <math>5x</math> which differ by a multiple of <math>2\pi</math> from either of these two solutions,
-		+
-		+
-	<math>~~\text{5}x=\text{ }\frac{\pi }{4}+2n\pi </math>~~	+
-	~~and~~	+
-	~~<math>\text{5}x=\frac{3\pi }{4}+2n~~\pi </math>,	+

-	~~where~~	+	{{Displayed math\|\|<math>5x = \frac{\pi}{4} + 2n\pi\qquad\text{and}\qquad 5x = \frac{3\pi}{4} + 2n\pi\,,</math>}}
-	<math>n</math>	+
-	~~is an arbitrary integer.~~	+

-	~~If we divide both of these by~~	+	where ''n'' is an arbitrary integer.
-	~~<math>\text{5}</math>, we obtain the solutions expressed in terms of x alone:~~	+

		+	If we divide both of these by 5, we obtain the solutions expressed in terms of ''x'' alone,

-	<math>x=\frac{\pi }{20}+\frac{2}{5}n\pi ~~</math>~~	+	{{Displayed math\|\|<math>x = \frac{\pi}{20} + \frac{2}{5}n\pi\qquad\text{and}\qquad x = \frac{3\pi}{20} + \frac{2}{5}n\pi\,,</math>}}
-	and	+
-	~~<math>~~x=\frac{3\pi }{20}+\frac{2}{5}n\pi </math>,	+

-	where	+	where ''n'' is an arbitrary integer.
-	~~<math>~~n~~</math>~~	+
-	is an arbitrary integer.	+

Ian at 13:55, 30 September 2008

2008-09-30T13:55:34Z

←Older revision		Revision as of 13:55, 30 September 2008
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-	{{~~NAVCONTENT_START~~}}	+	Apart from the fact that there is a
-	<~~center~~> ~~[[Image:4_4_2d-1(~~2~~).gif]]~~ </~~center~~>	+	<math>\text{5}x</math>, this is a normal trigonometric equation of the type
-	{{~~NAVCONTENT_STOP~~}}	+	<math>\text{sin }y\text{ }=a</math> . If we are only interested in solutions which satisfy
-	{~~{NAVCONTENT_START~~}}	+	<math>0\le \text{5}x\le \text{2}\pi </math>, then a sketch of the unit circle shows that there are two such solutions,
-	~~<center> [[Image:4_4_2d-2(2).gif]]~~ </~~center~~>	+	<math>\text{5}x=\text{ }\frac{\pi }{4}</math>
-	{{~~NAVCONTENT_STOP~~}}	+	and the reflectionally symmetric solution
		+	<math>\text{5}x=\text{ }\pi \text{-}\frac{\pi }{4}=\frac{3\pi }{4}</math>.
		+

	[[Image:4_4_2_d.gif\|center]]		[[Image:4_4_2_d.gif\|center]]
		+
		+	All of the equation's solutions are obtained from all values of 5x which differ by a multiple of 2π from either of these two solutions:
		+
		+
		+	<math>\text{5}x=\text{ }\frac{\pi }{4}+2n\pi </math>
		+	and
		+	<math>\text{5}x=\frac{3\pi }{4}+2n\pi </math>,
		+
		+	where
		+	<math>n</math>
		+	is an arbitrary integer.
		+
		+	If we divide both of these by
		+	<math>\text{5}</math>, we obtain the solutions expressed in terms of x alone:
		+
		+
		+	<math>x=\frac{\pi }{20}+\frac{2}{5}n\pi </math>
		+	and
		+	<math>x=\frac{3\pi }{20}+\frac{2}{5}n\pi </math>,
		+
		+	where
		+	<math>n</math>
		+	is an arbitrary integer.

Tekbot: Lösning 4.4:2d moved to Solution 4.4:2d: Robot: moved page

2008-09-09T10:03:03Z

Lösning 4.4:2d moved to Solution 4.4:2d: Robot: moved page

←Older revision

Revision as of 10:03, 9 September 2008

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

2008-08-21T07:18:39Z

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

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Madde at 12:16, 19 August 2008

2008-08-19T12:16:41Z

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2008-04-03T12:27:54Z

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