Solution 4.2:3e - Revision history

Tek at 08:07, 9 October 2008

Tek — Thu, 09 Oct 2008 08:07:31 GMT

←Older revision		Revision as of 08:07, 9 October 2008
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-	If we draw the line which has angle	+	If we draw the line which has angle <math>3\pi/4</math> relative to the positive ''x''-axis, it will cut the unit circle in the second quadrant.
-	<math>~~\text{~~3}{\pi }/{4~~}\;~~</math>	+
-	relative to the positive	+
-	~~<math>~~x~~</math>~~	+
-	-axis, it will cut the unit circle in the second quadrant.	+

	[[Image:4_2_3_e1.gif\|center]]		[[Image:4_2_3_e1.gif\|center]]
-	At first sight, the figure gives information about the signs of	+
-	<math>\cos {3\pi }/{4~~}\;~~</math>	+	At first sight, the figure gives information about the signs of <math>\cos (3\pi/4)</math> and <math>\sin (3\pi/4)</math>: cosine, which is the ''x''-coordinate for the intersection point, is negative, whereas sine, which is the ''y''-coordinate for the intersection point, is positive.
-	and	+
-	<math>\sin {3\pi }/{4~~}\;~~</math>: cosine, which is the	+
-	~~<math>~~x~~</math>~~	+
-	-coordinate for the intersection point, is negative, whereas sine, which is the	+
-	~~<math>~~y~~</math>~~	+
-	-coordinate for the intersection point, is positive.	+

	In order to go further and determine the coordinates of the intersection points, we concentrate on the second quadrant and introduce an auxiliary triangle which has the line associated with the angle as the hypotenuse and its other edges parallel to the coordinate axes.		In order to go further and determine the coordinates of the intersection points, we concentrate on the second quadrant and introduce an auxiliary triangle which has the line associated with the angle as the hypotenuse and its other edges parallel to the coordinate axes.
-

	[[Image:4_2_3_e_2.gif\|center]]		[[Image:4_2_3_e_2.gif\|center]]

-	In this triangle, we see that the angle	+	In this triangle, we see that the angle <math>\alpha</math> between the hypotenuse and the ''y''-axis is the part of the angle <math>3\pi/4</math> that lies in the second quadrant, i.e. <math>\alpha = 3\pi/4 - \pi/2 = \pi/4\,</math>. With the help of trigonometry, we can now determine the triangle's sides.
-	<math>\alpha </math>	+
-	between the hypotenuse and the	+
-	~~<math>~~y~~</math>~~	+
-	-axis is the part of the	+
-	<math>y</math>	+
-	~~-axis~~ that lies in the second quadrant, i.e.	+
-	<math>\alpha =~~\frac{~~3\pi }{4}-~~\frac{~~\pi }{2}=~~\frac{~~\pi }{4}</math>. With the help of trigonometry, we can now determine the triangle's sides.	+
-		+
-		+

-	[[Image:4_2_3_e3.gif\|center]]	+	{\| width="100%"
		+	\|width="50%" align="center"\|[[Image:4_2_3_e3.gif\|center]]
		+	\|width="50%" align="left"\|<math>\begin{align}\text{opposite} &= 1\cdot\sin\frac{\pi}{4} = \frac{1}{\sqrt{2}}\\[5pt] \text{adjacent} &= 1\cdot\cos\frac{\pi}{4} = \frac{1}{\sqrt{2}}\end{align}</math>
		+	\|}

	This shows that the intersection point has coordinates		This shows that the intersection point has coordinates
-	<math>~~\left~~( -\~~frac{1}{~~\sqrt{2}~~} \right.~~,\~~left. \frac{1}{~~\sqrt{2}~~} \right~~)</math>	+	<math>(-1/\!\sqrt{2}, 1/\!\sqrt{2})</math> (note the minus sign in the ''x''-coordinate) and therefore that <math>\sin (3\pi/4) = 1/\!\sqrt{2}\,</math>.
-	(note the minus sign in the	+
-	~~<math>~~x~~</math>~~	+
-	-coordinate) and therefore that	+
-	<math>\sin ~~\frac{~~3\pi }{4}=~~\frac{~~1}{\sqrt{2}}</math>.	+

Ian at 12:25, 28 September 2008

Ian — Sun, 28 Sep 2008 12:25:33 GMT

←Older revision		Revision as of 12:25, 28 September 2008
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-	{{~~NAVCONTENT_START~~}}	+	If we draw the line which has angle
-	~~<center> [[Image:4_2_3e-1(2).gif]]~~ </~~center~~>	+	<math>\text{3}{\pi }/{4}\;</math>
-	~~{{NAVCONTENT_STOP}}~~	+	relative to the positive
-	~~{{NAVCONTENT_START}}~~	+	<math>x</math>
-	<~~center~~> ~~[[Image:4_2_3e-2(2).gif]]~~ </~~center~~>	+	-axis, it will cut the unit circle in the second quadrant.
-	~~{{NAVCONTENT_STOP}}~~	+

	[[Image:4_2_3_e1.gif\|center]]		[[Image:4_2_3_e1.gif\|center]]
		+	At first sight, the figure gives information about the signs of
		+	<math>\cos {3\pi }/{4}\;</math>
		+	and
		+	<math>\sin {3\pi }/{4}\;</math>: cosine, which is the
		+	<math>x</math>
		+	-coordinate for the intersection point, is negative, whereas sine, which is the
		+	<math>y</math>
		+	-coordinate for the intersection point, is positive.
		+
		+	In order to go further and determine the coordinates of the intersection points, we concentrate on the second quadrant and introduce an auxiliary triangle which has the line associated with the angle as the hypotenuse and its other edges parallel to the coordinate axes.
		+
		+
	[[Image:4_2_3_e_2.gif\|center]]		[[Image:4_2_3_e_2.gif\|center]]
		+
		+	In this triangle, we see that the angle
		+	<math>\alpha </math>
		+	between the hypotenuse and the
		+	<math>y</math>
		+	-axis is the part of the
		+	<math>y</math>
		+	-axis that lies in the second quadrant, i.e.
		+	<math>\alpha =\frac{3\pi }{4}-\frac{\pi }{2}=\frac{\pi }{4}</math>. With the help of trigonometry, we can now determine the triangle's sides.
		+
		+
		+
	[[Image:4_2_3_e3.gif\|center]]		[[Image:4_2_3_e3.gif\|center]]
		+
		+	This shows that the intersection point has coordinates
		+	<math>\left( -\frac{1}{\sqrt{2}} \right.,\left. \frac{1}{\sqrt{2}} \right)</math>
		+	(note the minus sign in the
		+	<math>x</math>
		+	-coordinate) and therefore that
		+	<math>\sin \frac{3\pi }{4}=\frac{1}{\sqrt{2}}</math>.

Tekbot: Lösning 4.2:3e moved to Solution 4.2:3e: Robot: moved page

Tekbot — Tue, 09 Sep 2008 09:45:03 GMT

Lösning 4.2:3e moved to Solution 4.2:3e: Robot: moved page

←Older revision

Revision as of 09:45, 9 September 2008

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

Tekbot — Thu, 21 Aug 2008 07:09:39 GMT

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

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Madde at 14:41, 15 August 2008

Madde — Fri, 15 Aug 2008 14:41:49 GMT

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-	hej
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	[[Bild:4_2_3_e_2.gif\|center]]		[[Bild:4_2_3_e_2.gif\|center]]
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Madde at 14:39, 15 August 2008

Madde — Fri, 15 Aug 2008 14:39:57 GMT

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Madde at 14:39, 15 August 2008

Madde — Fri, 15 Aug 2008 14:39:40 GMT

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Madde at 14:37, 15 August 2008

Madde — Fri, 15 Aug 2008 14:37:18 GMT

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-	[[Bild:~~4_2_3_e3~~.gif\|center]]	+
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Madde at 14:30, 15 August 2008

Madde — Fri, 15 Aug 2008 14:30:29 GMT

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Oskar: Ny sida: {{NAVCONTENT_START}} Bild:4_2_3e-1(2).gif {{NAVCONTENT_STOP}} {{NAVCONTENT_START}} Bild:4_2_3e-2(2).gif {{NAVCONTENT_STOP}}

Oskar — Thu, 03 Apr 2008 07:55:53 GMT

Ny sida: {{NAVCONTENT_START}} <center> Bild:4_2_3e-1(2).gif </center> {{NAVCONTENT_STOP}} {{NAVCONTENT_START}} <center> Bild:4_2_3e-2(2).gif </center> {{NAVCONTENT_STOP}}

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-	[[Bild:~~4_2_3_e3~~.gif\|center]]	+
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