4.1 Angles and circles - Revision history

Jdinglis at 17:31, 10 November 2008

2008-11-10T17:31:02Z

Jdinglis at 17:25, 10 November 2008

2008-11-10T17:25:27Z

←Older revision		Revision as of 17:25, 10 November 2008
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	*To convert between degrees, radians and revolutions.		*To convert between degrees, radians and revolutions.
	*To calculate the area and circumference of sectors of a circle.		*To calculate the area and circumference of sectors of a circle.
-	*The ~~concepts~~ of right-angled triangles ~~including its legs and hypotenuse~~.	+	*The features of right-angled triangles.
	*To formulate and use the Pythagorean theorem.		*To formulate and use the Pythagorean theorem.
	*To calculate the distance between two points in the plane.		*To calculate the distance between two points in the plane.
-	*To sketch circles by completing the square ~~in their equations~~.	+	*To sketch circles by completing the square.
-	*The concepts of unit circle, tangent, radius, diameter, circumference, chord and arc.	+	*The concepts of the unit circle, tangent, radius, diameter, circumference, chord and arc.
	*To solve geometric problems that contain circles.		*To solve geometric problems that contain circles.
	}}		}}
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	[[Image:Gradskiva - 57°.gif\|\|center]]		[[Image:Gradskiva - 57°.gif\|\|center]]

-	*'''Radians.''' Another way to measure an angle, is to use the length of the arc ~~which subtends~~ the angle in relation to the radius ~~as a measure of the angle~~. This unit is called radian. A revolution is <math>2\pi</math> radians as the circumference of a circle is <math>2\pi r</math>, where <math>r</math> is the radius of the circle.	+	*'''Radians.''' Another way to measure an angle, is to use the length of the arc described by the angle in relation to the radius. This unit is called radian. A revolution is <math>2\pi</math> radians, since the circumference of a circle is <math>2\pi r</math>, where <math>r</math> is the radius of the circle.

	[[Image:Gradskiva - Radianer.gif\|\|center]]		[[Image:Gradskiva - Radianer.gif\|\|center]]
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	= \frac{\pi}{6}\ \mbox{ radians }</math></li>		= \frac{\pi}{6}\ \mbox{ radians }</math></li>
	<li><math>\frac{\pi}{8}\ \mbox { radians }		<li><math>\frac{\pi}{8}\ \mbox { radians }
-	= \frac{\pi}{8} \cdot (1 \; \mbox{~~radians~~}\,)	+	= \frac{\pi}{8} \cdot (1 \; \mbox{radian}\,)
	= \frac{\pi}{8} \cdot \frac{180^\circ}{\pi}		= \frac{\pi}{8} \cdot \frac{180^\circ}{\pi}
-	= 22{,}5^\circ</math></li>	+	= 22\mbox{.}5^\circ</math></li>
	</ol>		</ol>

	</div>		</div>

-	In some contexts, it may be useful to talk about negative angles and angles greater than 360°. This means that the same ~~direction~~ can be designated by different angles that differ from each other by an integral number of revolutions.	+	In some contexts, it may be useful to talk about negative angles and angles greater than 360°. This means that the same point on the circle can be designated by different angles that differ from each other by an integral number of revolutions.

	<center>{{:4.1 - Figure - Angles 45°, -315° and 405°}}</center>		<center>{{:4.1 - Figure - Angles 45°, -315° and 405°}}</center>
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	<ol type="a">		<ol type="a">
	<li> The angles <math>-55^\circ</math> and <math>665^\circ		<li> The angles <math>-55^\circ</math> and <math>665^\circ
-	</math> indicate the same ~~direction~~ because	+	</math> indicate the same point on the circle because
	{{Displayed math\|\|<math>		{{Displayed math\|\|<math>
	-55^\circ + 2 \cdot 360^\circ = 665^\circ\,\mbox{.}</math>}}</li>		-55^\circ + 2 \cdot 360^\circ = 665^\circ\,\mbox{.}</math>}}</li>

	<li> The angles <math>\frac{3\pi}{7}</math> and <math>		<li> The angles <math>\frac{3\pi}{7}</math> and <math>
-	-\frac{11\pi}{7}</math> indicate the same ~~direction~~ because	+	-\frac{11\pi}{7}</math> indicate the same point on the circle because
	{{Displayed math\|\|<math>		{{Displayed math\|\|<math>
	\frac{3\pi}{7} - 2\pi = -\frac{11\pi}{7}\,\mbox{.}</math>}}</li>		\frac{3\pi}{7} - 2\pi = -\frac{11\pi}{7}\,\mbox{.}</math>}}</li>

	<li> The angles <math>36^\circ</math> and <math>		<li> The angles <math>36^\circ</math> and <math>
-	216^\circ</math> do not specify the ~~same direction~~, but opposite ~~directions~~ since	+	216^\circ</math> do not specify the samepoint on the circle, but opposite points since
	{{Displayed math\|\|<math>		{{Displayed math\|\|<math>
	36^\circ + 180^\circ = 216^\circ\,\mbox{.}</math>}}</li>		36^\circ + 180^\circ = 216^\circ\,\mbox{.}</math>}}</li>
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	== Formula for distance in the plane ==		== Formula for distance in the plane ==

-	The theorem of Pythagoras is one of the most famous theorems in mathematics and says that in a right-angled triangle with the legs <math>a</math> and <math>b</math>, and ~~the~~ hypotenuse <math>c</math> then	+	The theorem of Pythagoras is one of the most famous theorems in mathematics and says that in a right-angled triangle with the legs <math>a</math> and <math>b</math>, and hypotenuse <math>c</math> then <math>a^2 + b^2 = c^2</math>.

	<div class="regel">		<div class="regel">
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	{\| width="100%"		{\| width="100%"
-	\|width="100%"\| ~~The~~ triangle on the right is	+	\|width="100%"\| Consider the triangle on the right. Then
	{{Displayed math\|\|<math>c^2= 3^2 + 4^2 = 9 +16 = 25</math>}}		{{Displayed math\|\|<math>c^2= 3^2 + 4^2 = 9 +16 = 25</math>}}
-	and therefore hypotenuse <math>c</math> is equal to	+	and therefore the hypotenuse <math>c</math> is equal to
	{{Displayed math\|\|<math>c=\sqrt{25} = 5\,\mbox{.}</math>}}		{{Displayed math\|\|<math>c=\sqrt{25} = 5\,\mbox{.}</math>}}
	\|align="right"\|{{:4.1 - Figure - A right-angled triangle with sides 3, 4 and 5}}		\|align="right"\|{{:4.1 - Figure - A right-angled triangle with sides 3, 4 and 5}}
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-	The distance <math>r</math> is called the ~~circle´s~~ radius and the point <math>(a,b)</math> is its centre. The figure below shows the other important concepts.	+	The distance <math>r</math> is called the circle's radius and the point <math>(a,b)</math> is its centre. The figure below shows the other important concepts.

	{\| align="center"		{\| align="center"
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	\|align="center"\| Arc of a circle		\|align="center"\| Arc of a circle
	\|\|		\|\|
-	\|align="center"\| ~~circumference~~	+	\|align="center"\| Circumference
	\|\|		\|\|
-	\|align="center"\| ~~sector~~ of a circle	+	\|align="center"\| Sector of a circle
	\|\|		\|\|
-	\|align="center"\|~~segment~~ of a circle	+	\|align="center"\|Segment of a circle
	\|}		\|}

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	{\| width="100%"		{\| width="100%"
-	\|\|A sector of a circle is ~~given~~ in the figure on the right.	+	\|\|A sector of a circle is shown in the figure on the right.
	<ol type="a">		<ol type="a">
	<li> Determine its arc length .		<li> Determine its arc length .
	<br>		<br>
	<br>		<br>
-	The central angle <math>50^\circ</math> ~~is in~~ radians	+	The central angle is <math>50^\circ</math>. In radians this is
	{{Displayed math\|\|<math>		{{Displayed math\|\|<math>
	50^\circ = 50 \cdot 1^\circ		50^\circ = 50 \cdot 1^\circ
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	\|}		\|}
	<ol style="list-style-type:none; padding-top:0; margin-top:0;">		<ol style="list-style-type:none; padding-top:0; margin-top:0;">
-	<li>The way radians have been defined means that the arc length is the radius multiplied by the angle measured in radians,	+	<li>The way radians have been defined means that the arc length is the radius multiplied by the angle measured in radians, so that the arc length is
	{{Displayed math\|\|<math>		{{Displayed math\|\|<math>
	3 \cdot \frac{5\pi}{18}\ \mbox{units }		3 \cdot \frac{5\pi}{18}\ \mbox{units }
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	<br>		<br>
	<br>		<br>
-	The ~~circle segment´s~~ share of the entire circle is	+	The segment's share of the entire circle is
	{{Displayed math\|\|<math>		{{Displayed math\|\|<math>
	\frac{50^\circ}{360^\circ} = \frac{5}{36}</math>}}		\frac{50^\circ}{360^\circ} = \frac{5}{36}</math>}}
-	and this means that its area is <math>\frac{5}{36}</math> parts of the circle area, which is <math>\pi r^2 = \pi 3^2 = 9\pi</math>~~, i.e~~.	+	and this means that its area is <math>\frac{5}{36}</math> parts of the circle area, which is <math>\pi r^2 = \pi 3^2 = 9\pi</math>. Hence the area is
	{{Displayed math\|\|<math>		{{Displayed math\|\|<math>
	\frac{5}{36} \cdot 9\pi\ \mbox{ units }= \frac{5\pi}{4}\ \mbox{ units. }</math>}}</li>		\frac{5}{36} \cdot 9\pi\ \mbox{ units }= \frac{5\pi}{4}\ \mbox{ units. }</math>}}</li>
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	</div>		</div>

-	A point <math>(x,y)</math> lies on the circle that has its centre at <math>(a,b)</math> and radius <math>r</math>, if its distance from the centre is equal to <math>r</math>. This condition can be formulated with the distance formula as	+	A point <math>(x,y)</math> lies on the circle that has its centre at <math>(a,b)</math> and radius <math>r</math>, if its distance from the centre is equal to <math>r</math>. This condition can be formulated with the distance formula.
	<div class="regel">		<div class="regel">
	{\| width="100%"		{\| width="100%"
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	\|width="100%"\|		\|width="100%"\|
	<ol type="a">		<ol type="a">
-	<li><math>(x-1)^2 + (y-2)^2 = 9\quad</math> is the equation for a circle with ~~its~~ centre at <math>(1,2)</math> and radius <math>\sqrt{9} = 3</math>.</li>	+	<li><math>(x-1)^2 + (y-2)^2 = 9\quad</math> is the equation for a circle with centre <math>(1,2)</math> and radius <math>\sqrt{9} = 3</math>.</li>
	</ol>		</ol>
	\|align="right"\|{{:4.1 - Figure - The equation (x - 1)² + (y - 2)² = 9}}		\|align="right"\|{{:4.1 - Figure - The equation (x - 1)² + (y - 2)² = 9}}
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	\|width="100%"\|		\|width="100%"\|
	<ol type="a" start=2>		<ol type="a" start=2>
-	<li><math>x^2 + (y-1)^2 = 1\quad</math> can be written as <math>(x-0)^2 + (y-1)^2 = 1</math> and is the equation of a circle with ~~its~~ centre at <math>(0,1)</math> and ~~having a~~ radius <math>\sqrt{1} = 1</math>.</li>	+	<li><math>x^2 + (y-1)^2 = 1\quad</math> can be written as <math>(x-0)^2 + (y-1)^2 = 1</math> and is the equation of a circle with centre <math>(0,1)</math> and radius <math>\sqrt{1} = 1</math>.</li>
	</ol>		</ol>
	\|align="right"\|{{:4.1 - Figure - The equation x² + (y - 1)² = 1}}		\|align="right"\|{{:4.1 - Figure - The equation x² + (y - 1)² = 1}}
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	\|width="100%"\|		\|width="100%"\|
	<ol type="a" start=3>		<ol type="a" start=3>
-	<li><math>(x+1)^2 + (y-3)^2 = 5\quad</math> can be written as <math>(x-(-1))^2 + (y-3)^2 = 5</math> and is the equation of a circle with ~~its~~ centre at <math>(-1,3)</math> and ~~having a~~ radius <math>\sqrt{5} \approx 2\textrm{.}236</math>.</li>	+	<li><math>(x+1)^2 + (y-3)^2 = 5\quad</math> can be written as <math>(x-(-1))^2 + (y-3)^2 = 5</math> and is the equation of a circle with centre <math>(-1,3)</math> and radius <math>\sqrt{5} \approx 2\textrm{.}236</math>.</li>
	</ol>		</ol>
	\|align="right"\|{{:4.1 - Figure - The equation (x + 1)² + (y - 3)² = 5}}		\|align="right"\|{{:4.1 - Figure - The equation (x + 1)² + (y - 3)² = 5}}

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	In some contexts, it may be useful to talk about negative angles and angles greater than 360°. This means that the same direction can be designated by different angles that differ from each other by an integral number of revolutions.		In some contexts, it may be useful to talk about negative angles and angles greater than 360°. This means that the same direction can be designated by different angles that differ from each other by an integral number of revolutions.

-	<center>{{:4.1 - ~~Figur~~ - ~~Vinklarna~~ 45°, -315° ~~och~~ 405°}}</center>	+	<center>{{:4.1 - Figure - Angles 45°, -315° and 405°}}</center>

	<div class="exempel">		<div class="exempel">

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	\|align="center" valign="bottom"\|{{:4.1 - Figure - The chord of a circle}}		\|align="center" valign="bottom"\|{{:4.1 - Figure - The chord of a circle}}
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-	\|align="center" valign="bottom"\|{{:4.1 - ~~Figur~~ - ~~Sekant~~}}	+	\|align="center" valign="bottom"\|{{:4.1 - Figure - Secant}}
	\|-		\|-
	\|align="center"\|Diameter		\|align="center"\|Diameter

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	We can interpret this as follows: The centre is at <math>(1,-2)</math> and the radius is <math>\sqrt{4}= 2</math>.		We can interpret this as follows: The centre is at <math>(1,-2)</math> and the radius is <math>\sqrt{4}= 2</math>.

-	<center>{{:4.1 - ~~Figur~~ - ~~Ekvationen~~ x² + y² - 2x + 4y + 1 = 0}}</center>	+	<center>{{:4.1 - Figure - The equation x² + y² - 2x + 4y + 1 = 0}}</center>
	</div>		</div>

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	<li><math>x^2 + (y-1)^2 = 1\quad</math> can be written as <math>(x-0)^2 + (y-1)^2 = 1</math> and is the equation of a circle with its centre at <math>(0,1)</math> and having a radius <math>\sqrt{1} = 1</math>.</li>		<li><math>x^2 + (y-1)^2 = 1\quad</math> can be written as <math>(x-0)^2 + (y-1)^2 = 1</math> and is the equation of a circle with its centre at <math>(0,1)</math> and having a radius <math>\sqrt{1} = 1</math>.</li>
	</ol>		</ol>
-	\|align="right"\|{{:4.1 - ~~Figur~~ - ~~Ekvationen~~ x² + (y - 1)² = 1}}	+	\|align="right"\|{{:4.1 - Figure - The equation x² + (y - 1)² = 1}}
	\|-		\|-
	\|width="100%"\|		\|width="100%"\|

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	\end{align*}</math>}}		\end{align*}</math>}}
	Since the point satisfies the circle equation it lies on the circle.		Since the point satisfies the circle equation it lies on the circle.
-	<center>{{:4.1 - ~~Figur~~ - ~~Ekvationen~~ (x - 4)² + y² = 13}}</center></li>	+	<center>{{:4.1 - Figure - The equation (x - 4)² + y² = 13}}</center></li>

	<li> Determine the equation for the circle that has its centre at <math>(3,4)</math> and goes through the point <math>(1,0)</math>.		<li> Determine the equation for the circle that has its centre at <math>(3,4)</math> and goes through the point <math>(1,0)</math>.

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	The circle equation is therefore		The circle equation is therefore
	{{Displayed math\|\|<math>(x-3)^2 + (y-4)^2 = 20 \; \mbox{.}</math>}}		{{Displayed math\|\|<math>(x-3)^2 + (y-4)^2 = 20 \; \mbox{.}</math>}}
-	<center>{{:4.1 - ~~Figur~~ - ~~Ekvationen~~ (x - 3)² + (y - 4)² = 20}}</center></li>	+	<center>{{:4.1 - Figure - The equation (x - 3)² + (y - 4)² = 20}}</center></li>
	</ol>		</ol>
	</div>		</div>

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	\|align="center" valign="bottom"\|{{:4.1 - Figure - Diameter}}		\|align="center" valign="bottom"\|{{:4.1 - Figure - Diameter}}
	\|width="15px"\|		\|width="15px"\|
-	\|align="center" valign="bottom"\|{{:4.1 - ~~Figur~~ - Tangent}}	+	\|align="center" valign="bottom"\|{{:4.1 - Figure - Tangent}}
	\|width="15px"\|		\|width="15px"\|
	\|align="center" valign="bottom"\|{{:4.1 - Figure - The chord of a circle}}		\|align="center" valign="bottom"\|{{:4.1 - Figure - The chord of a circle}}

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	{\| align="center"		{\| align="center"
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	\|width="15px"\|		\|width="15px"\|
	\|align="center" valign="bottom"\|{{:4.1 - Figur - Tangent}}		\|align="center" valign="bottom"\|{{:4.1 - Figur - Tangent}}

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	\|align="center" valign="bottom"\|{{:4.1 - Figure - The chord of a circle}}		\|align="center" valign="bottom"\|{{:4.1 - Figure - The chord of a circle}}
	\|width="15px"\|		\|width="15px"\|
-	\|align="center" valign="bottom"\|{{:4.1 - ~~Figur~~ - ~~Sekant~~}}	+	\|align="center" valign="bottom"\|{{:4.1 - Figure - Secant}}
	\|-		\|-
	\|align="center"\|Diameter		\|align="center"\|Diameter

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	<br>		<br>
	<br>		<br>
-	Since the point <math>(1,0)</math> lies on the circle, the radius of the circle must be equal to the distance of the point from <math>(1,0)</math> to the centre <math>(3,4)</math>. The distance formula ~~gives~~ that this distance is	+	Since the point <math>(1,0)</math> lies on the circle, the radius of the circle must be equal to the distance of the point from <math>(1,0)</math> to the centre <math>(3,4)</math>. The distance formula allows us to calculate that this distance is
	{{Displayed math\|\|<math>		{{Displayed math\|\|<math>
	c = \sqrt{(3-1)^2 + (4-0)^2} = \sqrt{4 +16} = \sqrt{20} \, \mbox{.}</math>}}		c = \sqrt{(3-1)^2 + (4-0)^2} = \sqrt{4 +16} = \sqrt{20} \, \mbox{.}</math>}}
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-	Let us try to write the equation ~~of the circle~~ in the form	+	Let us try to write the equation in the form
	{{Displayed math\|\|<math>(x – a)^2 + (y – b)^2 = r^2</math>}}		{{Displayed math\|\|<math>(x – a)^2 + (y – b)^2 = r^2</math>}}
-	because then we can directly read from this that the ~~midpoint~~ is <math>(a,b)</math> and the radius is <math>r</math>.	+	because then we can directly read from this that the centre is <math>(a,b)</math> and the radius is <math>r</math>.

	Start by completing the square for the terms containing <math>x</math> on the left-hand side		Start by completing the square for the terms containing <math>x</math> on the left-hand side
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	\underline{x^2-2x\vphantom{(}} + y^2+4y + 1		\underline{x^2-2x\vphantom{(}} + y^2+4y + 1
	= \underline{(x-1)^2-1^2} + y^2+4y + 1</math>}}		= \underline{(x-1)^2-1^2} + y^2+4y + 1</math>}}
-	~~(the~~ underlined terms shows the terms involved).	+	The underlined terms shows the terms involved.

-	~~Complete~~ the square for the terms containing <math>y</math>	+	Now complete the square for the terms containing <math>y</math>
	{{Displayed math\|\|<math>		{{Displayed math\|\|<math>
	(x-1)^2-1^2 + \underline{y^2+4y} + 1		(x-1)^2-1^2 + \underline{y^2+4y} + 1
	= (x-1)^2-1^2 + \underline{(y+2)^2-2^2} + 1\,\mbox{.}</math>}}		= (x-1)^2-1^2 + \underline{(y+2)^2-2^2} + 1\,\mbox{.}</math>}}

-	The left-hand side is equal to	+	The left-hand side is therefore equal to
-	{{Displayed math\|\|<math> (x-1)^2 + (y+2)^2-4 </math>}}	+	{{Displayed math\|\|<math> (x-1)^2 + (y+2)^2-4 </math>.}}

-	~~and moving over the 4 to to the right-hand side~~ we ~~get~~ the equation	+	Rearranging we see that the original equation is equivalent to
	{{Displayed math\|\|<math> (x-1)^2 + (y+2)^2 = 4 \, \mbox{.}</math>}}		{{Displayed math\|\|<math> (x-1)^2 + (y+2)^2 = 4 \, \mbox{.}</math>}}

-	~~We can interpret~~ this ~~as follows: The~~ centre is at <math>(1,-2)</math> and the radius is <math>\sqrt{4}= 2</math>.	+	From this we can easily see that the centre is <math>(1,-2)</math> and the radius is <math>\sqrt{4}= 2</math>.

	<center>{{:4.1 - Figure - The equation x² + y² - 2x + 4y + 1 = 0}}</center>		<center>{{:4.1 - Figure - The equation x² + y² - 2x + 4y + 1 = 0}}</center>
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-	'''Keep in mind that:'''	+	'''Keep in mind that...'''


	'''Reviews'''		'''Reviews'''

-	For those of you who want to deepen your ~~studies~~ or need more detailed explanations consider the following references:	+	For those of you who want to deepen your understanding or need more detailed explanations consider the following references:

-	[http://sv.wikipedia.org/wiki/Pythagoras_sats Learn more about Pythagoras theorem ~~in English~~ Wikipedia ]	+	[http://sv.wikipedia.org/wiki/Pythagoras_sats Learn more about Pythagoras theorem from Wikipedia ]

-	[http://mathworld.wolfram.com/Circle.html Read more ~~in Mathworld~~ about the circle ]	+	[http://mathworld.wolfram.com/Circle.html Read more about the circle on the Mathworld website]