4.2 Trigonometric functions - Revision history

Phil at 16:37, 6 January 2009

2009-01-06T16:37:51Z

←Older revision		Revision as of 16:37, 6 January 2009
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	</center>		</center>

-	The value of the ratio <math>\frac{a}{b}</math> is not dependent on the size of the triangle, but only on the angle <math>u</math>. For different values of the angle, you can get the value of the tangent either from a trigonometric table or by using a calculator (the ~~relevent~~ button is usually named tan).	+	The value of the ratio <math>\frac{a}{b}</math> is not dependent on the size of the triangle, but only on the angle <math>u</math>. For different values of the angle, you can get the value of the tangent either from a trigonometric table or by using a calculator (the relevant button is usually named tan).

	<div class="exempel">		<div class="exempel">
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	<center>{{:4.2 - Figure - Flagpole}}</center>		<center>{{:4.2 - Figure - Flagpole}}</center>

-	The flagpole and its shadow form a ~~rectangular~~ triangle where the vertical side is unknown (marked with <math>x</math> below).	+	The flagpole and its shadow form a right-angled triangle where the vertical side is unknown (marked with <math>x</math> below).

	<center>{{:4.2 - Figure - Flagpole triangle}}</center>		<center>{{:4.2 - Figure - Flagpole triangle}}</center>
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	and since <math>\tan 40^\circ \approx 0\textrm{.}84</math> we get		and since <math>\tan 40^\circ \approx 0\textrm{.}84</math> we get
	{{Displayed math\|\|<math>		{{Displayed math\|\|<math>
-	x = 5\,\mbox{m} \~~cdot~~ \tan 40^\circ \approx 5\,\mbox{m} \~~cdot~~ 0\textrm{.}84	+	x = 5\,\mbox{m} \times \tan 40^\circ \approx 5\,\mbox{m} \times 0\textrm{.}84
	= 4\textrm{.}2\,\mbox{m}\,\mbox{.}</math>}}		= 4\textrm{.}2\,\mbox{m}\,\mbox{.}</math>}}

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	{{Displayed math\|\|<math>\frac{22}{40} = \frac{x}{60}</math>}}		{{Displayed math\|\|<math>\frac{22}{40} = \frac{x}{60}</math>}}

-	which leads to <math>x=60 \~~cdot~~ \displaystyle \frac{22}{40} = 33</math>.	+	which leads to <math>x=60 \times \displaystyle \frac{22}{40} = 33</math>.
	</div>		</div>

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	{{Displayed math\|\|<math>\sin 38^\circ = \frac{x}{5}</math>,}}		{{Displayed math\|\|<math>\sin 38^\circ = \frac{x}{5}</math>,}}
	and if we know that <math>\sin 38^\circ \approx 0\textrm{.}616</math> then		and if we know that <math>\sin 38^\circ \approx 0\textrm{.}616</math> then
-	{{Displayed math\|\|<math>x = 5 \~~cdot~~ \sin 38^\circ \approx 5 \~~cdot~~ 0\textrm{.}616 \approx 3\textrm{.}1\,\mbox{.}</math>}}	+	{{Displayed math\|\|<math>x = 5 \times \sin 38^\circ \approx 5 \times 0\textrm{.}616 \approx 3\textrm{.}1\,\mbox{.}</math>}}
	\|-		\|-
	\| height="10px" \|		\| height="10px" \|
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	<center>{{:4.2 - Figure - A right-angled triangle with angle u and sides ½ and 1}}</center>		<center>{{:4.2 - Figure - A right-angled triangle with angle u and sides ½ and 1}}</center>

-	With the help of ~~the Pythagorean~~ theorem the side on the right can be determined:	+	With the help of Pythagoras' theorem the side on the right can be determined:

	<center>		<center>
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-	Using ~~the Pythagorean~~ theorem, we can determine the length <math>x</math> of the diagonal:	+	Using Pythagoras' theorem, we can determine the length <math>x</math> of the diagonal:
	{{Displayed math\|\|<math>		{{Displayed math\|\|<math>
	x^2 = 1^2 + 1^2		x^2 = 1^2 + 1^2
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	{{Displayed math\|\|<math>\tan u = \displaystyle\frac{\sin u}{\cos u}</math>}}		{{Displayed math\|\|<math>\tan u = \displaystyle\frac{\sin u}{\cos u}</math>}}

-	and the value of the tangent can be interpreted as the ~~slope for~~ the ~~radial line~~.	+	and the value of the tangent can be interpreted as the gradient of the radius.

Jdinglis at 17:58, 10 November 2008

2008-11-10T17:58:15Z

←Older revision		Revision as of 17:58, 10 November 2008
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	For those of you who want to deepen your understanding 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://dooku.miun.se/per.edstrom/interaktiv_matematik/trigonometri/cos_even.html Learn more about trigonometry on the "Interactive Mathematics" website]

	[http://en.wikipedia.org/wiki/Trigonometric_function Learn more about trigonometry from Wikipedia]		[http://en.wikipedia.org/wiki/Trigonometric_function Learn more about trigonometry from Wikipedia]

Jdinglis at 17:54, 10 November 2008

2008-11-10T17:54:59Z

←Older revision		Revision as of 17:54, 10 November 2008
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	'''Learning outcomes:'''		'''Learning outcomes:'''

-	After this section, you will have learned :	+	After this section, you will have learned:
	*The definition of acute, obtuse and right angles.		*The definition of acute, obtuse and right angles.
	*The definition of cosine, sine and tangent.		*The definition of cosine, sine and tangent.

Jdinglis at 17:54, 10 November 2008

2008-11-10T17:54:35Z

←Older revision		Revision as of 17:54, 10 November 2008
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	== Trigonometric functions for general angles ==		== Trigonometric functions for general angles ==

-	For angles of less than 0° or greater than 90° the trigonometric functions are defined using the unit circle (that is the circle that has its centre at the origin and has a radius 1).	+	For angles less than 0° or greater than 90° the trigonometric functions are defined using the unit circle (that is the circle that has its centre at the origin and has radius 1).

	<div class="regel">		<div class="regel">
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	\|-		\|-
	\| width="90%" valign="center"\|		\| width="90%" valign="center"\|
-	The trigonometric functions <math>\cos u</math> and <math>\sin u</math> are ''x''- and ''y''- coordinates of the ~~intersection between~~ the unit circle ~~and the radial line that forms~~ the angle <math>u</math> ~~with~~ the ~~positive ''x''-axis~~.	+	The trigonometric functions <math>\cos u</math> and <math>\sin u</math> are the ''x''- and ''y''- coordinates of the point on the unit circle reached by turning through the angle <math>u</math>, as shown in the diagram on the right.
	\| width="10%" \|		\| width="10%" \|
	\| align="right" valign="center" \|		\| align="right" valign="center" \|
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	</div>		</div>

-	~~Tangent~~ function is defined as	+	The tangent function is then defined as

	{{Displayed math\|\|<math>\tan u = \displaystyle\frac{\sin u}{\cos u}</math>}}		{{Displayed math\|\|<math>\tan u = \displaystyle\frac{\sin u}{\cos u}</math>}}
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	\| width="90%" align="left" valign="center" \|		\| width="90%" align="left" valign="center" \|
	<math>\begin{align*}		<math>\begin{align*}
-	\cos 104^\circ &\approx -0{,}24\\[8pt]	+	\cos 104^\circ &\approx -0\mbox{.}24\\[8pt]
-	\sin 104^\circ &\approx 0{,}97\\[8pt]	+	\sin 104^\circ &\approx 0\mbox{.}97\\[8pt]
-	\tan 104^\circ &\approx \dfrac{0{,}97}{-0{,}24} \approx -4{,}0\\	+	\tan 104^\circ &\approx \dfrac{0\mbox{.}97}{-0\mbox{.}24} \approx -4\mbox{.}0\\
	\end{align*}</math>		\end{align*}</math>
	\|-		\|-
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	\| width="90%" align="left" valign="center" \|		\| width="90%" align="left" valign="center" \|
	<math>\begin{align*}		<math>\begin{align*}
-	\cos 201^\circ &\approx -0{,}93\\[8pt]	+	\cos 201^\circ &\approx -0\mbox{.}93\\[8pt]
-	\sin 201^\circ &\approx -0{,}36\\[8pt]	+	\sin 201^\circ &\approx -0\mbox{.}36\\[8pt]
-	\tan 201^\circ &\approx \dfrac{-0{,}36}{-0{,}93} \approx 0{,}4\\	+	\tan 201^\circ &\approx \dfrac{-0\mbox{.}36}{-0\mbox{.}93} \approx 0\mbox{.}4\\
	\end{align*}</math>		\end{align*}</math>
	\|}		\|}
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	<br>		<br>
	<br>		<br>
-	By drawing angle<math>-40^\circ</math> in the unit circle one obtains a radial line which has a negative slope, ~~i.e.~~ <math>\tan (-40^\circ)</math> is negative. </li>	+	By drawing angle <math>-40^\circ</math> in the unit circle one obtains a radial line which has a negative slope, so that <math>\tan (-40^\circ)</math> is negative. </li>
	</ol>		</ol>
	\| width="5%" \|		\| width="5%" \|
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	<br>		<br>
	<br>		<br>
-	~~Rewriting~~	+	Note that
	{{Displayed math\|\|<math>		{{Displayed math\|\|<math>
	\frac{2\pi}{3} = \frac{4\pi}{6}		\frac{2\pi}{3} = \frac{4\pi}{6}
	= \frac{3\pi+ \pi}{6}		= \frac{3\pi+ \pi}{6}
-	= \frac{\pi}{2} + \frac{\pi}{6}</math>}}	+	= \frac{\pi}{2} + \frac{\pi}{6}</math>.}}

-	shows that the angle <math>2\pi/3</math> ~~lands~~ in the the second quadrant and makes the angle <math>\pi/6</math> with the positive ''y''-axis. If we draw an extra triangle as in the figure below on the right, we see that the <math>2\pi/3</math>- point on the unit circle has a ''y''-coordinate, which is equal to the adjacent side <math>\cos \frac{\pi}{6} = \sqrt{3}/2</math>. So we have that	+	This shows that the point on the unit circle corresponding to the angle <math>2\pi/3</math> is in the the second quadrant and makes the angle <math>\pi/6</math> with the positive ''y''-axis. If we draw an extra triangle as in the figure below on the right, we see that the <math>2\pi/3</math>- point on the unit circle has a ''y''-coordinate which is equal to the adjacent side <math>\cos \frac{\pi}{6} = \sqrt{3}/2</math>. So we have that
	{{Displayed math\|\|<math>		{{Displayed math\|\|<math>
	\sin\frac{2\pi}{3} = \frac{\sqrt{3}}{2}\,\mbox{.}</math>}}		\sin\frac{2\pi}{3} = \frac{\sqrt{3}}{2}\,\mbox{.}</math>}}
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-	== ~~The~~ trigonometric functions ~~graphs~~ ==	+	== Graphs of the trigonometric functions ==

-	In the last section, we used a unit circle to define cosine and sine of arbitrary angles and we ~~often~~ will use the unit circle in the future, for example, to derive trigonometric relationships and solve trigonometric equations. However, there are certain characteristics of the trigonometric functions that are better illustrated by drawing their graphs.	+	In the last section we used a unit circle to define the cosine and sine of arbitrary angles, and we will often use the unit circle in the future, for example, to derive trigonometric relationships and solve trigonometric equations. However, there are certain characteristics of the trigonometric functions that are better illustrated by drawing their graphs.


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	In these graphs, we might observe several things more clearly than in the unit circle. Some examples are:		In these graphs, we might observe several things more clearly than in the unit circle. Some examples are:

-	* The curves for cosine and sine repeat themselves after a change in angle of <math>2\pi</math>, that is ~~the~~ <math>\cos (x+2\pi) = \cos x</math> and <math>\sin (x+2\pi) = \sin x</math>. ~~For~~ the unit circle <math>2\pi</math> corresponds to a revolution, and after a complete revolution ~~angles~~ return to the same ~~location~~ on the ~~unit~~ circle ~~and therefore have the same coordinates~~.	+	* The curves for cosine and sine repeat themselves after a change in angle of <math>2\pi</math>, that is <math>\cos (x+2\pi) = \cos x</math> and <math>\sin (x+2\pi) = \sin x</math>. To see why this is true, note that on the unit circle <math>2\pi</math> corresponds to a complete revolution, and after a complete revolution we return to the same point on the circle.

	*The curve for the tangent repeats itself after a change in angle of <math>\pi</math>, that is <math>\tan (x+\pi) = \tan x</math>. Two angles which differ by <math>\pi</math> share the same line through the origin of the unit circle and thus their radial lines have the same slope.		*The curve for the tangent repeats itself after a change in angle of <math>\pi</math>, that is <math>\tan (x+\pi) = \tan x</math>. Two angles which differ by <math>\pi</math> share the same line through the origin of the unit circle and thus their radial lines have the same slope.
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-	'''Keep in mind that: '''	+	'''Keep in mind that...'''

	If you have studied trigonometry, then you should not be afraid to use it in geometric problems. It often produces a simpler solution.		If you have studied trigonometry, then you should not be afraid to use it in geometric problems. It often produces a simpler solution.

-	You may need to spend a lot of time on understanding how to use a unit circle to define the trigonometric functions.	+	You may need to spend a lot of time understanding how to use a unit circle to define the trigonometric functions.

-	~~Get~~ into the habit of calculating with precise trigonometric values. It ~~provides a~~ good training in calculating fractions and eventually algebraic rational expressions.	+	You should get into the habit of calculating with precise trigonometric values. It is good training in calculating fractions and will eventually help you handle algebraic rational expressions.

	'''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://dooku.miun.se/per.edstrom/interaktiv_matematik/trigonometri/cos_even.html Learn more about trigonometry ~~in Per Edström~~ "Interactive Mathematics"]	+	[http://dooku.miun.se/per.edstrom/interaktiv_matematik/trigonometri/cos_even.html Learn more about trigonometry on the "Interactive Mathematics" website]

-	[http://en.wikipedia.org/wiki/Trigonometric_function Learn more about trigonometry ~~in the English~~ Wikipedia]	+	[http://en.wikipedia.org/wiki/Trigonometric_function Learn more about trigonometry from Wikipedia]

-	[http://en.wikipedia.org/wiki/Unit_circle Learn more about the unit circle ~~in the English~~ Wikipedia]	+	[http://en.wikipedia.org/wiki/Unit_circle Learn more about the unit circle from Wikipedia]


	'''Useful web sites'''		'''Useful web sites'''

-	[http://www.math.kth.se/online/images/sinus_och_cosinus_i_enhetscirkeln.swf Experiment with the sine and cosine ~~in the unit circle~~]	+	[http://www.math.kth.se/online/images/sinus_och_cosinus_i_enhetscirkeln.swf Experiment with the sine and cosine functions]

	[http://www.math.psu.edu/dlittle/java/geometry/euclidean/toolbox.html Experiment with Euclidean geometry]		[http://www.math.psu.edu/dlittle/java/geometry/euclidean/toolbox.html Experiment with Euclidean geometry]
	</div>		</div>

Jdinglis at 17:41, 10 November 2008

2008-11-10T17:41:20Z

←Older revision		Revision as of 17:41, 10 November 2008
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	== Trigonometry of right-angled triangles ==		== Trigonometry of right-angled triangles ==

-	In the right-angled triangle below the ratio between the length <math>a</math> of the side opposite the angle and the length <math>b</math> of the adjacent side is called the tangent of the angle <math>u</math> and is written as <math>\tan u</math>.	+	In the right-angled triangle below, the ratio between the length <math>a</math> of the side opposite the angle and the length <math>b</math> of the adjacent side is called the tangent of the angle <math>u</math>, and is written as <math>\tan u</math>.

	<center>		<center>
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	</center>		</center>

-	The value of the ratio <math>\frac{a}{b}</math> is not dependent on the size of the triangle, but only on the angle <math>u</math>. For different values of the angle, you can get the ~~equivalent~~ value of the tangent either from a trigonometric table or by using a calculator (the relevent button is usually named tan).	+	The value of the ratio <math>\frac{a}{b}</math> is not dependent on the size of the triangle, but only on the angle <math>u</math>. For different values of the angle, you can get the value of the tangent either from a trigonometric table or by using a calculator (the relevent button is usually named tan).

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

-	There are two other ratios in right-angled triangles that have special names~~, and one~~ is <math>\cos u = b/c</math> ("cosine of <math>u</math>") and the ~~other~~ <math>\sin u = a/c</math> (" sine of <math>u</math>").	+	There are two other ratios in right-angled triangles that have special names. The first is <math>\cos u = b/c</math> ("cosine of <math>u</math>") and the second is <math>\sin u = a/c</math> (" sine of <math>u</math>").

	<center>		<center>
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	\| width="10%" \|		\| width="10%" \|
	\| width="85%" align="left" valign="top" \|		\| width="85%" align="left" valign="top" \|
-	~~The~~ definition of sine ~~gives that~~	+	From the definition of sine we have
-	{{Displayed math\|\|<math>\sin 38^\circ = \frac{x}{5}</math>}}	+	{{Displayed math\|\|<math>\sin 38^\circ = \frac{x}{5}</math>,}}
-	and if we know that <math>\sin 38^\circ \approx 0\textrm{.}616</math> then ~~we get~~	+	and if we know that <math>\sin 38^\circ \approx 0\textrm{.}616</math> then
	{{Displayed math\|\|<math>x = 5 \cdot \sin 38^\circ \approx 5 \cdot 0\textrm{.}616 \approx 3\textrm{.}1\,\mbox{.}</math>}}		{{Displayed math\|\|<math>x = 5 \cdot \sin 38^\circ \approx 5 \cdot 0\textrm{.}616 \approx 3\textrm{.}1\,\mbox{.}</math>}}
	\|-		\|-
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	\| width="10%" \|		\| width="10%" \|
	\| width="85%" align="left" valign="top" \|		\| width="85%" align="left" valign="top" \|
-	Cosine is the ratio between the adjacent side and the hypotenuse	+	Cosine is the ratio between the adjacent side and the hypotenuse, so
	{{Displayed math\|\|<math>\cos 34^\circ = \frac{3}{x}\,\mbox{.}</math>}}		{{Displayed math\|\|<math>\cos 34^\circ = \frac{3}{x}\,\mbox{.}</math>}}
	Thus		Thus
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	<center>{{:4.2 - Figure - A right-angled triangle with angle u and sides ½ and 1}}</center>		<center>{{:4.2 - Figure - A right-angled triangle with angle u and sides ½ and 1}}</center>

-	With the help of the Pythagorean theorem the side on the right can be determined	+	With the help of the Pythagorean theorem the side on the right can be determined:

	<center>		<center>
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	== Some standard angles ==		== Some standard angles ==

-	For some angles namely 30°, 45° and 60° it is relatively easy to calculate the exact values of the trigonometric functions.	+	For some angles, namely 30°, 45° and 60°, it is relatively easy to calculate the exact values of the trigonometric functions.

	<div class="exempel">		<div class="exempel">
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-	Using the Pythagorean theorem, we can determine the length <math>x</math> of the diagonal,	+	Using the Pythagorean theorem, we can determine the length <math>x</math> of the diagonal:
	{{Displayed math\|\|<math>		{{Displayed math\|\|<math>
	x^2 = 1^2 + 1^2		x^2 = 1^2 + 1^2
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	x = \sqrt{1^2 + 1^2} = \sqrt{2}\,\mbox{.}</math>}}		x = \sqrt{1^2 + 1^2} = \sqrt{2}\,\mbox{.}</math>}}

-	Each triangle has the diagonal as the hypotenuse~~, thus~~ we can obtain the value of the trigonometric functions for the angle <math>45^\circ</math>.	+	Each triangle has the diagonal as the hypotenuse. Thus we can obtain the value of the trigonometric functions for the angle <math>45^\circ</math>:


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-	The Pythagorean theorem ~~shows~~ that the vertical side of ~~either~~ half-triangle is <math>x=\sqrt{3}/2</math>. ~~From one of these half-triangles~~ we get that	+	The Pythagorean theorem allows us to calculate that the length of the vertical side of half-triangle is <math>x=\sqrt{3}/2</math>. Using the definitions we then get that


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	\| align="left" valign="center" \|		\| align="left" valign="center" \|
	<math>\begin{align*}		<math>\begin{align*}
-	\cos 30^\circ &= \frac{\sqrt{3}/2}{1} = \frac{\sqrt{3}}{2}\,;\\[8pt]	+	\cos 30^\circ &= \frac{\sqrt{3}/2}{1} = \frac{\sqrt{3}}{2}\,\\[8pt]
-	\sin 30^\circ &= \frac{1/2}{1} = \frac{1}{2}\,;\\[8pt]	+	\sin 30^\circ &= \frac{1/2}{1} = \frac{1}{2}\,\\[8pt]
-	\tan 30^\circ &= \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}}\,;\\	+	\tan 30^\circ &= \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}}\,\\
	\end{align*}		\end{align*}
	\qquad\quad		\qquad\quad

Jdinglis at 17:34, 10 November 2008

2008-11-10T17:34:47Z

←Older revision		Revision as of 17:34, 10 November 2008
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	After this section, you will have learned :		After this section, you will have learned :
-	*The ~~concepts~~ of acute, obtuse and right angles.	+	*The definition of acute, obtuse and right angles.
-	*The definition of cosine, sine and tangent ~~in the unit circle~~.	+	*The definition of cosine, sine and tangent.
	*The values of cosine, sine and tangent for the standard angles <math>0</math>, <math>\pi/6</math> , <math>\pi/4</math> , <math>\pi/3</math> and <math>\pi/2</math> by heart.		*The values of cosine, sine and tangent for the standard angles <math>0</math>, <math>\pi/6</math> , <math>\pi/4</math> , <math>\pi/3</math> and <math>\pi/2</math> by heart.
-	*To determine the values of cosine, sine and tangent of arguments that can be reduced to a standard angle ~~in a quadrant of the unit circle~~.	+	*To determine the values of cosine, sine and tangent of arguments that can be reduced to a standard angle.
	* To sketch graphs of cosine, sine and tangent.		* To sketch graphs of cosine, sine and tangent.
	*To solve trigonometric problems involving right-angled triangles.		*To solve trigonometric problems involving right-angled triangles.
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	== Trigonometry of right-angled triangles ==		== Trigonometry of right-angled triangles ==

-	In the right-angled triangle below the ratio between the ~~opposite side~~ <math>a</math> ~~and~~ the ~~adjacent~~ side <math>b</math> is called the tangent of the angle <math>u</math> and is written as <math>\tan u</math>.	+	In the right-angled triangle below the ratio between the length <math>a</math> of the side opposite the angle and the length <math>b</math> of the adjacent side is called the tangent of the angle <math>u</math> and is written as <math>\tan u</math>.

	<center>		<center>

Tek: Change name of included figure

2008-09-16T08:07:49Z

Change name of included figure

←Older revision		Revision as of 08:07, 16 September 2008
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	Determine <math>\sin u</math> in the triangle		Determine <math>\sin u</math> in the triangle

-	<center>{{:4.2 - Figure - A right-angled triangle with angle ~~20°~~ and sides ½ and 1}}</center>	+	<center>{{:4.2 - Figure - A right-angled triangle with angle u and sides ½ and 1}}</center>

	With the help of the Pythagorean theorem the side on the right can be determined		With the help of the Pythagorean theorem the side on the right can be determined

Tekbot: Robot: Automated text replacement (-{{:4.2 - Figur - Rätvinklig triangel med vinkeln u och sidorna a, b och c}} +{{:4.2 - Figure - A right-angled triangle with angle u and sides a, b and c}})

2008-09-15T14:54:39Z

Robot: Automated text replacement (-{{:4.2 - Figur - Rätvinklig triangel med vinkeln u och sidorna a, b och c}} +{{:4.2 - Figure - A right-angled triangle with angle u and sides a, b and c}})

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-	{{:4.2 - ~~Figur~~ - ~~Rätvinklig triangel med vinkeln~~ u ~~och sidorna~~ a, b ~~och~~ c}}	+	{{:4.2 - Figure - A right-angled triangle with angle u and sides a, b and c}}
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Tekbot: Robot: Automated text replacement (-{{:4.2 - Figur - Rätvinklig triangel med vinkeln u och sidor 3, 4 och 5}} +{{:4.2 - Figure - A right-angled triangle with angle u and sides 3, 4 and 5}})

2008-09-15T14:53:31Z

Robot: Automated text replacement (-{{:4.2 - Figur - Rätvinklig triangel med vinkeln u och sidor 3, 4 och 5}} +{{:4.2 - Figure - A right-angled triangle with angle u and sides 3, 4 and 5}})

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-	{{:4.2 - ~~Figur~~ - ~~Rätvinklig triangel med vinkeln~~ u ~~och sidor~~ 3, 4 ~~och~~ 5}}	+	{{:4.2 - Figure - A right-angled triangle with angle u and sides 3, 4 and 5}}
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Tekbot: Robot: Automated text replacement (-{{:4.2 - Figur - Rätvinklig triangel med vinkeln u och kateterna a och b}} +{{:4.2 - Figure - A right-angled triangle with angle u and legs a and b}})

2008-09-15T14:52:11Z

Robot: Automated text replacement (-{{:4.2 - Figur - Rätvinklig triangel med vinkeln u och kateterna a och b}} +{{:4.2 - Figure - A right-angled triangle with angle u and legs a and b}})

←Older revision		Revision as of 14:52, 15 September 2008
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-	{{:4.2 - ~~Figur~~ - ~~Rätvinklig triangel med vinkeln~~ u ~~och kateterna~~ a ~~och~~ b}}	+	{{:4.2 - Figure - A right-angled triangle with angle u and legs a and b}}
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←Older revision		Revision as of 14:53, 15 September 2008
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-	{{:4.2 - ~~Figur~~ - ~~Rätvinklig triangel med vinkeln~~ u ~~och sidor~~ 3, 4 ~~och~~ 5}}	+	{{:4.2 - Figure - A right-angled triangle with angle u and sides 3, 4 and 5}}
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