Solution 4.4:7c - Revision history

Tek at 07:59, 14 October 2008

Tek — Tue, 14 Oct 2008 07:59:06 GMT

←Older revision		Revision as of 07:59, 14 October 2008
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-	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 at 13:20, 1 October 2008

Ian — Wed, 01 Oct 2008 13:20:21 GMT

←Older revision		Revision as of 13:20, 1 October 2008
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-	{{~~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

Tekbot — Tue, 09 Sep 2008 10:08:23 GMT

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

←Older revision

Revision as of 10:08, 9 September 2008

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

Tekbot — Thu, 21 Aug 2008 07:21:19 GMT

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

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-	<center> [[~~Bild~~:4_4_7c-1(3).gif]] </center>	+	<center> [[Image:4_4_7c-1(3).gif]] </center>
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-	[[~~Bild~~:4_4_7_c2.gif\|center]]	+	[[Image:4_4_7_c2.gif\|center]]

Madde at 13:24, 19 August 2008

Madde — Tue, 19 Aug 2008 13:24:14 GMT

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Oskar — Thu, 03 Apr 2008 12:44:18 GMT

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