Solution 4.4:5c - Revision history

Tek at 14:09, 13 October 2008

Tek — Mon, 13 Oct 2008 14:09:51 GMT

←Older revision		Revision as of 14:09, 13 October 2008
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-	For a fixed value of	+	For a fixed value of ''u'', an equality of the form
-	~~<math>~~u~~</math>~~, an equality of the form	+

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

-	~~<math>\cos u=\cos v</math>~~	+	is satisfied by two angles ''v'' in the unit circle,
-		+
-		+
-	is satisfied by two angles	+
-	~~<math>~~v~~</math>~~	+
-	in the unit circle:	+
-		+
-		+
-	~~<math>v=u</math>~~	+
-	~~and~~	+
-	~~<math>v=-u</math>~~	+

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

	[[Image:4_4_5_c.gif\|center]]		[[Image:4_4_5_c.gif\|center]]

-	This means that all angles	+	This means that all angles ''v'' which satisfy the equality are
-	~~<math>~~v~~</math>~~	+
-	which satisfy the equality are	+

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

-	~~<math>v=u+2n\pi </math>~~	+	where ''n'' is an arbitrary integer.
-	~~and~~	+
-	~~<math>v=-u+2n\pi </math>~~	+
-		+
-		+
-	where	+
-	~~<math>~~n~~\text{ }</math>~~	+
-	is an arbitrary integer.	+

	Therefore, the equation		Therefore, the equation

-		+	{{Displayed math\|\|<math>\cos 5x=\cos (x+\pi/5)</math>}}
-	<math>\cos 5x=\cos ~~\left~~( x+{\pi }/{5~~}\; \right~~)</math>	+
-		+

	has the solutions		has the solutions

-		+	{{Displayed math\|\|<math>\left\{\begin{align} 5x&=x+\frac{\pi}{5}+2n\pi\quad\text{or}\\[5pt] 5x &= -x-\frac{\pi}{5}+2n\pi\,\textrm{.}\end{align}\right.</math>}}
-	<math>5x=x+\frac{\pi }{5}+2n\pi ~~</math>~~	+
-	or	+
-		+
-	~~<math>~~5x=-x-\frac{\pi }{5}+2n\pi </math>	+

-	If we collect	+	If we collect ''x'' onto one side, we end up with
-	~~<math>~~x~~\text{ }</math>~~	+
-	onto one side, we end up with	+

		+	{{Displayed math\|\|<math>\left\{\begin{align}
		+	x &= \frac{\pi}{20} + \frac{n\pi}{2}\,,\\[5pt]
		+	x &= -\frac{\pi }{30}+\frac{n\pi}{3}\,,
		+	\end{align}\right.</math>}}

-	~~<math>\left\{ \begin{array}{*{35}l}~~	+	where ''n'' is an arbitrary integer.
-	~~x=\frac{\pi }{20}+\frac{1}{2}~~n~~\pi \\~~	+
-	~~x=-\frac{\pi }{30}+\frac{1}{3}n\pi \\~~	+
-	~~\end{array} \right.</math>~~	+
-	(	+
-	~~<math>n\text{ }</math>~~	+
-	an arbitrary integer).	+

Ian at 11:18, 1 October 2008

Ian — Wed, 01 Oct 2008 11:18:33 GMT

←Older revision		Revision as of 11:18, 1 October 2008
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-	~~{{NAVCONTENT_START}}~~	+	For a fixed value of
-	<~~center~~> ~~[[Image:4_4_5c-1(2).gif]]~~ </~~center~~>	+	<math>u</math>, an equality of the form
-	~~{{NAVCONTENT_STOP}}~~	+
-	~~{{NAVCONTENT_START}}~~	+
-	<~~center~~> ~~[[Image~~:~~4_4_5c-2(2).gif]]~~ </~~center~~>	+	<math>\cos u=\cos v</math>
-	~~{{NAVCONTENT_STOP}}~~	+
		+
		+	is satisfied by two angles
		+	<math>v</math>
		+	in the unit circle:
		+
		+
		+	<math>v=u</math>
		+	and
		+	<math>v=-u</math>


	[[Image:4_4_5_c.gif\|center]]		[[Image:4_4_5_c.gif\|center]]
		+
		+	This means that all angles
		+	<math>v</math>
		+	which satisfy the equality are
		+
		+
		+	<math>v=u+2n\pi </math>
		+	and
		+	<math>v=-u+2n\pi </math>
		+
		+
		+	where
		+	<math>n\text{ }</math>
		+	is an arbitrary integer.
		+
		+	Therefore, the equation
		+
		+
		+	<math>\cos 5x=\cos \left( x+{\pi }/{5}\; \right)</math>
		+
		+
		+	has the solutions
		+
		+
		+	<math>5x=x+\frac{\pi }{5}+2n\pi </math>
		+	or
		+
		+	<math>5x=-x-\frac{\pi }{5}+2n\pi </math>
		+
		+	If we collect
		+	<math>x\text{ }</math>
		+	onto one side, we end up with
		+
		+
		+	<math>\left\{ \begin{array}{*{35}l}
		+	x=\frac{\pi }{20}+\frac{1}{2}n\pi \\
		+	x=-\frac{\pi }{30}+\frac{1}{3}n\pi \\
		+	\end{array} \right.</math>
		+	(
		+	<math>n\text{ }</math>
		+	an arbitrary integer).

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

Tekbot — Tue, 09 Sep 2008 10:06:23 GMT

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

←Older revision

Revision as of 10:06, 9 September 2008

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

Tekbot — Thu, 21 Aug 2008 07:20:19 GMT

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

←Older revision		Revision as of 07:20, 21 August 2008
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	{{NAVCONTENT_START}}		{{NAVCONTENT_START}}
-	<center> [[~~Bild~~:4_4_5c-1(2).gif]] </center>	+	<center> [[Image:4_4_5c-1(2).gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}
	{{NAVCONTENT_START}}		{{NAVCONTENT_START}}
-	<center> [[~~Bild~~:4_4_5c-2(2).gif]] </center>	+	<center> [[Image:4_4_5c-2(2).gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}


-	[[~~Bild~~:4_4_5_c.gif\|center]]	+	[[Image:4_4_5_c.gif\|center]]

Madde at 13:12, 19 August 2008

Madde — Tue, 19 Aug 2008 13:12:17 GMT

←Older revision		Revision as of 13:12, 19 August 2008
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	<center> [[Bild:4_4_5c-2(2).gif]] </center>		<center> [[Bild:4_4_5c-2(2).gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}
		+
		+
		+	[[Bild:4_4_5_c.gif\|center]]

Oskar: Ny sida: {{NAVCONTENT_START}} Bild:4_4_5c-1(2).gif {{NAVCONTENT_STOP}} {{NAVCONTENT_START}} Bild:4_4_5c-2(2).gif {{NAVCONTENT_STOP}}

Oskar — Thu, 03 Apr 2008 12:37:23 GMT

Ny sida: {{NAVCONTENT_START}} <center> Bild:4_4_5c-1(2).gif </center> {{NAVCONTENT_STOP}} {{NAVCONTENT_START}} <center> Bild:4_4_5c-2(2).gif </center> {{NAVCONTENT_STOP}}

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