Solution 4.4:6c - Revision history

Tek at 06:55, 14 October 2008

Tek — Tue, 14 Oct 2008 06:55:12 GMT

←Older revision		Revision as of 06:55, 14 October 2008
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-	If we use the trigonometric relation	+	If we use the trigonometric relation <math>\sin (-x) = -\sin x</math>, the equation can be rewritten as
-	<math>\~~text{~~sin ~~}\left~~( -x ~~\right~~)=-\~~text{~~sin }x</math>, the equation can be rewritten as	+
-		+
-		+
-	~~<math>\sin 2x=\sin \left( -x \right)</math>~~	+

		+	{{Displayed math\|\|<math>\sin 2x = \sin (-x)\,\textrm{.}</math>}}

	In exercise 4.4:5a, we saw that an equality of the type		In exercise 4.4:5a, we saw that an equality of the type

-		+	{{Displayed math\|\|<math>\sin u = \sin v</math>}}
-	<math>\sin u=\sin v~~\quad~~ </math>	+
-		+

	is satisfied if		is satisfied if

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

-	~~<math>u=v+2n\pi </math>~~	+	where ''n'' is an arbitrary integer. The consequence of this is that the solutions to the equation satisfy
-	or	+
-	~~<math>u=\pi -v+2n\pi </math>~~	+
-		+
-		+
-	where	+
-	~~<math>~~n~~\text{ }</math>~~	+
-	is an arbitrary integer. The consequence of this is that the solutions to the equation satisfy	+
-		+
-		+
-	~~<math>2x=-x+2n\pi </math>~~	+
-	or	+
-	~~<math>2x=\pi -\left( -x \right)+2n\pi </math>~~	+

		+	{{Displayed math\|\|<math>2x = -x+2n\pi\qquad\text{or}\qquad 2x = \pi-(-x)+2n\pi\,,</math>}}

	i.e.		i.e.

		+	{{Displayed math\|\|<math>3x = 2n\pi\qquad\text{or}\qquad x = \pi +2n\pi\,\textrm{.}</math>}}

-	~~<math>3x=2n\pi </math>~~	+	The solutions to the equation are thus
-	or	+
-	~~<math>x=\pi +2n\pi </math>~~	+

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

-	~~The solutions to the equation are thus~~	+	where ''n'' is an arbitrary integer.
-		+
-		+
-	~~<math>\left\{ \begin{array}{*{35}l}~~	+
-	~~x=\frac{2n\pi }{3} \\~~	+
-	~~x=\pi +2n\pi \\~~	+
-	~~\end{array} \right.</math>~~	+
-	(	+
-	~~<math>~~n~~\text{ }</math>~~	+
-	an arbitrary integer)	+

Ian at 11:54, 1 October 2008

Ian — Wed, 01 Oct 2008 11:54:54 GMT

←Older revision		Revision as of 11:54, 1 October 2008
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-	{{~~NAVCONTENT_START}~~}	+	If we use the trigonometric relation
-	<~~center~~> ~~[[Image~~:~~4_4_6c~~.~~gif]]~~ </~~center~~>	+	<math>\text{sin }\left( -x \right)=-\text{sin }x</math>, the equation can be rewritten as
-	{{~~NAVCONTENT_STOP~~}}	+
		+
		+	<math>\sin 2x=\sin \left( -x \right)</math>
		+
		+
		+	In exercise 4.4:5a, we saw that an equality of the type
		+
		+
		+	<math>\sin u=\sin v\quad </math>
		+
		+
		+	is satisfied if
		+
		+
		+	<math>u=v+2n\pi </math>
		+	or
		+	<math>u=\pi -v+2n\pi </math>
		+
		+
		+	where
		+	<math>n\text{ }</math>
		+	is an arbitrary integer. The consequence of this is that the solutions to the equation satisfy
		+
		+
		+	<math>2x=-x+2n\pi </math>
		+	or
		+	<math>2x=\pi -\left( -x \right)+2n\pi </math>
		+
		+
		+	i.e.
		+
		+
		+	<math>3x=2n\pi </math>
		+	or
		+	<math>x=\pi +2n\pi </math>
		+
		+
		+	The solutions to the equation are thus
		+
		+
		+	<math>\left\{ \begin{array}{*{35}l}
		+	x=\frac{2n\pi }{3} \\
		+	x=\pi +2n\pi \\
		+	\end{array} \right.</math>
		+	(
		+	<math>n\text{ }</math>
		+	an arbitrary integer)

Tekbot: Lösning 4.4:6c moved to Solution 4.4:6c: Robot: moved page

Tekbot — Tue, 09 Sep 2008 10:07:23 GMT

Lösning 4.4:6c moved to Solution 4.4:6c: Robot: moved page

←Older revision

Revision as of 10:07, 9 September 2008

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

Tekbot — Thu, 21 Aug 2008 07:20:49 GMT

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

←Older revision		Revision as of 07:20, 21 August 2008
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-	<center> [[~~Bild~~:4_4_6c.gif]] </center>	+	<center> [[Image:4_4_6c.gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}

Oskar: Ny sida: {{NAVCONTENT_START}} Bild:4_4_6c.gif {{NAVCONTENT_STOP}}

Oskar — Thu, 03 Apr 2008 12:46:35 GMT

Ny sida: {{NAVCONTENT_START}} <center> Bild:4_4_6c.gif </center> {{NAVCONTENT_STOP}}

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