Solution 4.4:5a - Revision history

Tek at 13:53, 13 October 2008

Tek — Mon, 13 Oct 2008 13:53:18 GMT

←Older revision		Revision as of 13:53, 13 October 2008
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	If we consider for a moment the equality		If we consider for a moment the equality

		+	{{Displayed math\|\|<math>\sin u = \sin v</math>\|(*)}}

-	~~<math>\sin~~ u~~=\sin~~ v~~\quad \quad \quad (*)</math>~~	+	where ''u'' has a fixed value, there are usually two angles ''v'' in the unit circle which ensure that the equality holds,

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

		+	[[Image:4_4_5_a.gif\|\|center]]

-	~~where~~	+	(The only exception is when <math>u = \pi/2</math> or <math>u=3\pi/2</math>, in which case <math>u</math> and <math>\pi-u</math> correspond to the same direction and there is only one angle ''v'' which satisfies the equality.)
-	<math>u</math>	+
-	~~has a fixed value, there are usually two angles~~	+
-	<math>v</math>	+
-	in the ~~unit circle~~ which ~~ensure that~~ the equality ~~holds:~~	+

		+	We obtain all the angles ''v'' which satisfy (*) by adding multiples of <math>2\pi</math>,

-	<math>v=u~~</math>~~	+	{{Displayed math\|\|<math>v = u+2n\pi\qquad\text{and}\qquad v = \pi-u+2n\pi\,,</math>}}
-	and	+
-	~~<math>~~v=\pi -u</math>	+

-		+	where ''n'' is an arbitrary integer.
-	~~[[Image:4_4_5_a.gif]]~~	+
-		+
-		+
-	~~(The only exception is when~~	+
-	~~<math>u={\pi }/{2}\;\text{ }</math>~~	+
-	or	+
-	~~<math>u=3{\pi }/{2}\;\text{ }</math>, in which case~~	+
-	~~<math>u</math>~~	+
-	~~and~~	+
-	~~<math>\pi -u\text{ }</math>~~	+
-	~~correspond to the same direction and there is only one angle~~	+
-	~~<math>v</math>~~	+
-	~~which satisfies the equality.)~~	+
-		+
-	~~We obtain all the angles~~	+
-	~~<math>v</math>~~	+
-	~~which satisfy (*) by adding multiples of~~	+
-	~~<math>\text{2}\pi </math>,~~	+
-		+
-		+
-	~~<math>v=u+2n\pi </math>~~	+
-	~~and~~	+
-	~~<math>v=\pi -u+2n\pi </math>~~	+
-		+
-		+
-	where	+
-	~~<math>~~n~~\text{ }</math>~~	+
-	is an arbitrary integer.	+

	If we now go back to our equation		If we now go back to our equation

		+	{{Displayed math\|\|<math>\sin 3x = \sin x</math>}}

-	~~<math>\sin 3x=\sin x</math>~~	+	the reasoning above shows that the equation is only satisfied when
-		+
-		+
-	the reasoning above shows that the equation is only satisfied when	+
-		+
-		+
-	~~<math>3x=x+2n\pi </math>~~	+
-	or	+
-	~~<math>3x=\pi -x+2n\pi </math>~~	+
-		+

-	~~If we make~~	+	{{Displayed math\|\|<math>3x = x+2n\pi\qquad\text{or}\qquad 3x = \pi-x+2n\pi\,\textrm{.}</math>}}
-	<math>x</math>	+
-	~~the subject of each equation, we obtain the full solution to the equation:~~	+

		+	If we make ''x'' the subject of each equation, we obtain the full solution to the equation,

-	<math>\left\{ \begin{~~array}{*{35}l~~}	+	{{Displayed math\|\|<math>\left\{\begin{align}
-	x=0+n\pi \\	+	x &= 0+n\pi\,,\\[5pt]
-	x=\frac{\pi }{4}+\frac{1}{2}~~n\pi~~ \\	+	x &= \frac{\pi}{4}+\frac{n\pi}{2}\,\textrm{.}
-	\end{~~array~~} \right.</math>	+	\end{align}\right.</math>}}

Ian at 10:55, 1 October 2008

Ian — Wed, 01 Oct 2008 10:55:48 GMT

←Older revision		Revision as of 10:55, 1 October 2008
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-	~~{{NAVCONTENT_START}}~~	+	If we consider for a moment the equality
-	<~~center~~> ~~[[Image:4_4_5a-1~~(2)~~.gif]]~~ </~~center~~>	+
-	~~{{NAVCONTENT_STOP}}~~	+
-	~~{{NAVCONTENT_START}}~~	+	<math>\sin u=\sin v\quad \quad \quad (*)</math>
-	<~~center~~> ~~[[Image~~:~~4_4_5a~~-~~2(2).gif]]~~ </~~center~~>	+
-	~~{{NAVCONTENT_STOP}}~~	+
		+
		+	where
		+	<math>u</math>
		+	has a fixed value, there are usually two angles
		+	<math>v</math>
		+	in the unit circle which ensure that the equality holds:
		+
		+
		+	<math>v=u</math>
		+	and
		+	<math>v=\pi -u</math>
		+

	[[Image:4_4_5_a.gif]]		[[Image:4_4_5_a.gif]]
		+
		+
		+	(The only exception is when
		+	<math>u={\pi }/{2}\;\text{ }</math>
		+	or
		+	<math>u=3{\pi }/{2}\;\text{ }</math>, in which case
		+	<math>u</math>
		+	and
		+	<math>\pi -u\text{ }</math>
		+	correspond to the same direction and there is only one angle
		+	<math>v</math>
		+	which satisfies the equality.)
		+
		+	We obtain all the angles
		+	<math>v</math>
		+	which satisfy (*) by adding multiples of
		+	<math>\text{2}\pi </math>,
		+
		+
		+	<math>v=u+2n\pi </math>
		+	and
		+	<math>v=\pi -u+2n\pi </math>
		+
		+
		+	where
		+	<math>n\text{ }</math>
		+	is an arbitrary integer.
		+
		+	If we now go back to our equation
		+
		+
		+	<math>\sin 3x=\sin x</math>
		+
		+
		+	the reasoning above shows that the equation is only satisfied when
		+
		+
		+	<math>3x=x+2n\pi </math>
		+	or
		+	<math>3x=\pi -x+2n\pi </math>
		+
		+
		+	If we make
		+	<math>x</math>
		+	the subject of each equation, we obtain the full solution to the equation:
		+
		+
		+	<math>\left\{ \begin{array}{*{35}l}
		+	x=0+n\pi \\
		+	x=\frac{\pi }{4}+\frac{1}{2}n\pi \\
		+	\end{array} \right.</math>

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

Tekbot — Tue, 09 Sep 2008 10:05:43 GMT

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

←Older revision

Revision as of 10:05, 9 September 2008

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

Tekbot — Thu, 21 Aug 2008 07:19:59 GMT

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

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-	[[~~Bild~~:4_4_5_a.gif]]	+	[[Image:4_4_5_a.gif]]

Madde at 13:10, 19 August 2008

Madde — Tue, 19 Aug 2008 13:10:15 GMT

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Oskar — Thu, 03 Apr 2008 12:35:00 GMT

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