Aus Online Mathematik Brückenkurs 1

Version vom 16:47, 13. Jun. 2008 (bearbeiten) Tek (Diskussion | Beiträge) K (Regerate images and tabs) ← Zum vorherigen Versionsunterschied		Version vom 12:51, 16. Jul. 2008 (bearbeiten) (rückgängig) Ian (Diskussion | Beiträge) Zum nächsten Versionsunterschied →
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-	{{Vald flik|[[4.3 Trigonometriska samband|Teori]]}}	+	{{Vald flik|[[4.3 Trigonometriska samband|Theory]]}}
-	{{Ej vald flik|[[4.3 Övningar|Övningar]]}}	+	{{Ej vald flik|[[4.3 Övningar|Theory]]}}
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	{{Info|		{{Info|
-	'''Innehåll:'''	+	'''Content:'''
-	*Trigonometriska ettan	+	* Pythagorean identity
-	*Formeln för dubbla och halva vinkeln	+	* The double-angle and half-angle formulas
-	*Additions- och subtraktionsformlerna	+	* Addition and subtraction formulas
	}}		}}
	{{Info|		{{Info|
-	'''Lärandemål:'''	+	'''Learning outcome:'''
-	Efter detta avsnitt ska du ha lärt dig att:	+	After this section, you will have learned how to:
-	*Härleda trigonometriska samband från symmetrier i enhetscirkeln.	+	*Derive trigonometric relationships from symmetries in the unit circle.
-	*Förenkla trigonometriska uttryck med hjälp av de trigonometriska sambanden.	+	* Simplify trigonometric expressions with the help of trigonometric formulas.
	}}		}}
-	== Inledning ==	+	== Introduction ==
-	Det finns en mängd trigonometriska samband, med vilka man kan översätta mellan sinus-, cosinus- och tangensvärden för en vinkel eller multiplar av en vinkel. Dessa brukar också kallas trigonometriska identiteter, eftersom de endast är olika sätt att beskriva ett och samma uttryck med hjälp av olika trigonometriska funktioner. Här kommer vi att beskriva några av dessa trigonometriska samband. Det finns många fler än vi kan behandla här. De flesta kan härledas utifrån den s k '''trigonometriska ettan''' och additionsformlerna (se nedan), vilka är viktiga att kunna utantill.	+	There is a variety of trigonometric formulas to use if one wishes to transform between the sine, cosine or tangent for an angle or multiples of an angle. They are usually known as the trigonometric identities, since they only lead to different ways to describe a single expression using a variety of trigonometric functions. Here we will give some of these trigonometric relationships. There are many more than we can deal with in this course. Most can be derived from the so-called Pythagorean identity and the addition and subtraction formulas or identities (see below), which are important to know by heart.
-	== Trigonometriska ettan ==	+	== Pythagorean identity ==
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-	Detta samband är det mest grundläggande, men är i själva verket ingenting annat än Pythagoras sats, tillämpad i enhetscirkeln. Den rätvinkliga triangeln till höger visar att	+	This identity is the most basic, but is in fact nothing more than Pythagoras theorem, applied to the unit circle. The right-angled triangle on the right shows that
	{{Fristående formel||<math>(\sin v)^2 + (\cos v)^2 = 1\,\mbox{,}</math>}}		{{Fristående formel||<math>(\sin v)^2 + (\cos v)^2 = 1\,\mbox{,}</math>}}
-	vilket brukar skrivas <math>\sin^2\!v + \cos^2\!v = 1</math>.	+	which is usually written as <math>\sin^2\!v + \cos^2\!v = 1</math>.
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	{{:4.3 - Figur - Trigonometriska ettan}}		{{:4.3 - Figur - Trigonometriska ettan}}
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-	== Symmetrier ==	+	== Symmetries ==
-	Med hjälp av enhetscirkeln och spegling kan man tack vare de trigonometriska funktionernas symmetrier hitta en stor mängd samband mellan cosinus och sinus.	+	With the help of the unit circle and reflection, and exploiting the symmetries of the trigonometric functions one obtains a large amount of relationships between the cosine and sine functions.
	<div class="regel">		<div class="regel">
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-	Istället för att försöka lära sig alla dessa samband utantill kan det vara bättre att lära sig härleda dem i enhetscirkeln.	+	Instead of trying to learn all of these relationships by heart, it might be better to learn how to derive them from the unit circle.
-	'''Spegling i ''x''-axeln'''	+	'''Reflction in the ''x''-axis'''
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	|-		|-
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	<br>		<br>
-	När en vinkel <math>v</math> speglas i ''x''-axeln blir den <math>-v</math>.	+	When an angle <math>v</math> is reflected in the ''x''-axis it becomes<math>-v</math>.
-	Speglingen påverkar inte ''x''-koordinaten medan ''y''-koordinaten byter tecken	+	Reflction does not affect the ''x''- coordinate while the ''y''-oordinate changes sign.
	{{Fristående formel||<math>\begin{align*}		{{Fristående formel||<math>\begin{align*}
	\cos(-v) &= \cos v\,\mbox{,}\\		\cos(-v) &= \cos v\,\mbox{,}\\
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-	'''Spegling i ''y''-axeln'''	+	'''Reflction in the ''y''-axis'''
	{|		{|
	|-		|-
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	<br>		<br>
-	Vid spegling i ''y''-axeln ändras vinkeln <math>v</math> till <math>\pi-v</math> (spegelbilden bildar vinkeln <math>v</math> mot den negativa ''x''-axeln).	+	Reflection in the ''y''-axis changes the angle <math>v</math> to <math>\pi-v</math> ((the reflection makes an angle <math>v</math> with the negative ''x''-axis).
-	Speglingen påverkar inte ''y''-koordinaten medan ''x''-koordinaten byter tecken	+	Reflction does not affect the ''y''-coordinate while the ''x''-coordinate changes sign.
	{{Fristående formel||<math>\begin{align*}		{{Fristående formel||<math>\begin{align*}
	\cos(\pi-v) &= -\cos v\,\mbox{,}\\		\cos(\pi-v) &= -\cos v\,\mbox{,}\\
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-	'''Spegling i linjen ''y = x'' '''	+	''' Reflction in the line ''y = x'' '''
	{|		{|
	|-		|-
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	<br>		<br>
-	Vinkeln <math>v</math> ändras till vinkeln <math>\pi/2 - v</math> (spegelbilden bildar vinkeln <math>v</math> mot den positiva ''y''-axeln).	+	The angle <math>v</math> changes to <math>\pi/2 - v</math> ( the reflection makes an angle <math>v</math> with the positive ''y''-axis).
-	Speglingen gör att ''x''- och ''y''-koordinaterna byter plats	+	Reflction causes the ''x''- and ''y''-coordinates to change places
	{{Fristående formel||<math>\begin{align*}		{{Fristående formel||<math>\begin{align*}
	\cos \Bigl(\frac{\pi}{2} - v \Bigr) &= \sin v\,\mbox{.}\\		\cos \Bigl(\frac{\pi}{2} - v \Bigr) &= \sin v\,\mbox{.}\\
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-	'''Vridning med vinkeln <math>\mathbf{\pi/2}</math>'''	+	''' Rotation by an angle of <math>\mathbf{\pi/2}</math>'''
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	<br>		<br>
-	En vridning <math>\pi/2</math> av vinkeln <math>v</math> betyder att vinkeln blir <math>v+ \pi/2</math>.	+	A rotation <math>\pi/2</math> of the angle <math>v</math> means that the angle becomes <math>v+ \pi/2</math>.
-	Vridningen gör att ''x''-koordinaten blir ny ''y''-koordinat och ''y''-koordinaten blir ny ''x''-koordinat fast med omvänt tecken	+	The rotation turns the ''x''- coordinate into the new ''y''- coordinate and the ''y''- coordinates turns into the new ''x''-coordinate though with the opposite sign.
	{{Fristående formel||<math>\begin{align*}		{{Fristående formel||<math>\begin{align*}
	\cos \Bigl(v+\frac{\pi}{2}\Bigr) &= -\sin v\,\mbox{,}\\		\cos \Bigl(v+\frac{\pi}{2}\Bigr) &= -\sin v\,\mbox{,}\\
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-	Alternativt kan man få fram dessa samband genom att spegla och/eller förskjuta graferna. Om man exempelvis vill ha ett samband där <math>\cos v</math> uttrycks med hjälp av sinus så kan man förskjuta grafen för cosinus så att den passar med sinuskurvan. Detta kan göras på flera olika sätt, men mest naturligt faller det sig att skriva <math>\cos v = \sin (v + \pi / 2)</math>. För att undvika misstag kan man kontrollera att det stämmer för några olika värden på <math>v</math>.	+	Alternatively, one can get these relationships by reflecting and / or displacing graphs. For instance, if we want to have a formula in which <math>\cos v</math> uttrycks is expressed in terms of a sine one can displace the graph for cosine to fit the sine curve. This can be done in several ways, but the most natural is to write <math>\cos v = \sin (v + \pi / 2)</math>. To avoid mistakes, one can check that this is true for several different values of <math>v</math>.
	<center>{{:4.3 - Figur - Kurvorna y = cos x och y = sin x}}</center>		<center>{{:4.3 - Figur - Kurvorna y = cos x och y = sin x}}</center>
-	Kontroll: <math>\ \cos 0 = \sin (0 + \pi / 2)=1</math>.	+	Control: <math>\ \cos 0 = \sin (0 + \pi / 2)=1</math>.
-	== Additions- och subtraktionsformlerna och formler för dubbla vinkeln ==	+	== The addition and subtraction formulas and double-angle and half-angle formulas ==
-	Ofta behöver man behandla uttryck där två eller flera vinklar är inblandade, t.ex. <math>\sin(u+v)</math>. Man behöver då de s.k. additionsformlerna. För sinus och cosinus har formlerna utseendet	+	One often needs to deal with expressions in which two or more angles are involved, such as <math>\sin(u+v)</math>. One will then need the so-called " addition formulas “. For sine and cosine the formulas are
	<div class="regel">		<div class="regel">
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	</div>		</div>
-	Om man vill veta sinus eller cosinus för dubbla vinkeln, dvs <math>\sin 2v</math> eller <math>\cos 2v</math>, så kan man skriva uttrycken som <math>\sin(v + v)</math> eller <math>\cos(v + v)</math> och använda additionsformlerna ovan och få	+	If one wants to know the sine or cosine of a double angle, that is <math>\sin 2v</math> or <math>\cos 2v</math>, one can write these expressions as <math>\sin(v + v)</math> or <math>\cos(v + v)</math> and use the addition formulas above and get the double-angle
	<div class="regel">		<div class="regel">
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	</div>		</div>
-	Ur dessa samband kan vi sedan få fram formler för halva vinkeln. Genom att byta ut <math>2v</math> mot <math>v</math>, och följdaktligen <math>v</math> mot <math>v/2</math>, i formeln för <math>\cos 2v</math> får vi att	+	From these relationships, one can then get the formulas for half angles. By replacing <math>2v</math> by <math>v</math>, and consequently <math>v</math> by <math>v/2</math>, in the formula for <math>\cos 2v</math> one gets that
	{{Fristående formel||<math>		{{Fristående formel||<math>
	\cos v = \cos^2\!\frac{v}{2} – \sin^2\!\frac{v}{2}\,\mbox{.}</math>}}		\cos v = \cos^2\!\frac{v}{2} – \sin^2\!\frac{v}{2}\,\mbox{.}</math>}}
-	Vill vi ha en formel för <math>\sin(v/2)</math> så använder vi därefter den trigonometriska ettan för att bli av med <math>\cos^2(v/2)</math>	+	If we want a formula for <math>\sin(v/2)</math> we use the Pythagorean identity to get rid of <math>\cos^2(v/2)</math>
	{{Fristående formel||<math>		{{Fristående formel||<math>
	\cos v = 1 – \sin^2\!\frac{v}{2} – \sin^2\!\frac{v}{2}		\cos v = 1 – \sin^2\!\frac{v}{2} – \sin^2\!\frac{v}{2}
	= 1 – 2\sin^2\!\frac{v}{2}</math>}}		= 1 – 2\sin^2\!\frac{v}{2}</math>}}
-	dvs.	+	i.e.
	<div class="regel">		<div class="regel">
	{{Fristående formel||<math>		{{Fristående formel||<math>
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	</div>		</div>
-	På motsvarande sätt kan vi med den trigonometriska ettan göra oss av med <math>\sin^2(v/2)</math>. Då får vi istället	+	Similarly, we can use the Pythagorean identity to get rid of <math>\sin^2(v/2)</math>. Then we will have instead
	<div class="regel">		<div class="regel">
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-	[[4.3 Övningar|Övningar]]	+	[[4.3 Övningar|Exercises]]
	<div class="inforuta" style="width:580px;">		<div class="inforuta" style="width:580px;">
-	'''Råd för inläsning'''	+	'''Study advice'''
-	'''Grund- och slutprov'''	+	'''The basic and final tests''
-	Efter att du har läst texten och arbetat med övningarna ska du göra grund- och slutprovet för att bli godkänd på detta avsnitt. Du hittar länken till proven i din student lounge.	+	After you have read the text and worked through the exercises, you should do the basic and final tests to pass this section. You can find the link to the tests in your student lounge.
-	'''Tänk på att:'''	+	'''Keep in mind that:'''
-	Enhetscirkeln är ett ovärderligt hjälpmedel för att hitta trigonometriska samband. Sådana finns det gott om och det är ingen idé att försöka lära sig alla utantill. Det är också tidsödande att behöva slå upp och leta fram dem hela tiden. Därför är det mycket bättre att du lär dig använda enhetscirkeln.	+	The unit circle is an invaluable tool for finding trigonometric relationships. They are a multitude and there is no point in trying to learn all of them by heart. It is also time-consuming to have to look them up all the time. Therefore, it is much better that you learn how to use the unit circle.
-	Den allra mest kända trigonometriska formeln är den s k trigonometriska ettan. Den gäller för alla vinklar, inte bara för spetsiga. Den hänger ihop med Pythagoras sats.	+	The most famous trigonometric formula is the so-called Pythagorean identity. It applies to all angles, not just for acute angles It is based on the Pythagoras theorem.
-	'''Lästips'''
-	för dig som vill fördjupa dig ytterligare eller behöver en längre förklaring vill vi tipsa om:	+	'''Reviews'''
-	[http://www.theducation.se/kurser/umaprep/4_trigonometri/43_trig_formler/432_addisionsformlerna/index.asp Läs mer om trigonometriska formler i Theducations gymnasielexikon]	+	For those of you who want to deepen your studies or need more detailed explanations consider the following reference
-	[http://www.theducation.se/kurser/umaprep/4_trigonometri/43_trig_formler/432_addisionsformlerna/index.asp Läs mer om area-, sinus och cosinussatserna i Theducations gymnasielexikon]	+	[http://www.theducation.se/kurser/umaprep/4_trigonometri/43_trig_formler/432_addisionsformlerna/index.asp Learn more about trigonometric formulas in Theducations gymnasielexikon ]
-	[http://matmin.kevius.com/trigonometri.html Läs mer om trigonometri i Bruno Kevius matematiska ordlista]	+	[http://www.theducation.se/kurser/umaprep/4_trigonometri/43_trig_formler/432_addisionsformlerna/index.asp Learn more about areas, and the sine and cosine theorems in Theducations gymnasielexikon ]
		+
		+	[http://matmin.kevius.com/trigonometri.html Läs mer om trigonometri i Learn more about trigonometry in Bruno Kevius mathematical glossary ]
	'''Länktips'''		'''Länktips'''
-	[http://www.ies.co.jp/math/java/trig/cosbox/cosbox.html Experimentera med cosinus "lådan"]	+	[http://www.ies.co.jp/math/java/trig/cosbox/cosbox.html Experiment with the cosine “box” ]
-	[http://www.kth.se Testa dig själv i trigonometri - slå ditt eget ekord]	+	[http://www.kth.se Test yourself trigonometry - beat your own record]
	</div>		</div>

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Version vom 12:51, 16. Jul. 2008

Vorlage:Vald flik Vorlage:Ej vald flik

Introduction

There is a variety of trigonometric formulas to use if one wishes to transform between the sine, cosine or tangent for an angle or multiples of an angle. They are usually known as the trigonometric identities, since they only lead to different ways to describe a single expression using a variety of trigonometric functions. Here we will give some of these trigonometric relationships. There are many more than we can deal with in this course. Most can be derived from the so-called Pythagorean identity and the addition and subtraction formulas or identities (see below), which are important to know by heart.

Pythagorean identity

This identity is the most basic, but is in fact nothing more than Pythagoras theorem, applied to the unit circle. The right-angled triangle on the right shows that Vorlage:Fristående formel which is usually written as \displaystyle \sin^2\!v + \cos^2\!v = 1.

4.3 - Figur - Trigonometriska ettan

Symmetries

With the help of the unit circle and reflection, and exploiting the symmetries of the trigonometric functions one obtains a large amount of relationships between the cosine and sine functions.

Instead of trying to learn all of these relationships by heart, it might be better to learn how to derive them from the unit circle.

Reflction in the x-axis

4.3 - Figur - Spegling i x-axeln

When an angle \displaystyle v is reflected in the x-axis it becomes\displaystyle -v. Reflction does not affect the x- coordinate while the y-oordinate changes sign. Vorlage:Fristående formel

Reflction in the y-axis

4.3 - Figur - Spegling i y-axeln

Reflection in the y-axis changes the angle \displaystyle v to \displaystyle \pi-v ((the reflection makes an angle \displaystyle v with the negative x-axis). Reflction does not affect the y-coordinate while the x-coordinate changes sign. Vorlage:Fristående formel

Reflction in the line y = x

4.3 - Figur - Spegling i linjen y = x

The angle \displaystyle v changes to \displaystyle \pi/2 - v ( the reflection makes an angle \displaystyle v with the positive y-axis). Reflction causes the x- and y-coordinates to change places Vorlage:Fristående formel

Rotation by an angle of \displaystyle \mathbf{\pi/2}

4.3 - Figur - Vridning med vinkeln π/2

A rotation \displaystyle \pi/2 of the angle \displaystyle v means that the angle becomes \displaystyle v+ \pi/2. The rotation turns the x- coordinate into the new y- coordinate and the y- coordinates turns into the new x-coordinate though with the opposite sign. Vorlage:Fristående formel

Alternatively, one can get these relationships by reflecting and / or displacing graphs. For instance, if we want to have a formula in which \displaystyle \cos v uttrycks is expressed in terms of a sine one can displace the graph for cosine to fit the sine curve. This can be done in several ways, but the most natural is to write \displaystyle \cos v = \sin (v + \pi / 2). To avoid mistakes, one can check that this is true for several different values of \displaystyle v.

4.3 - Figur - Kurvorna y = cos x och y = sin x

Control: \displaystyle \ \cos 0 = \sin (0 + \pi / 2)=1.

The addition and subtraction formulas and double-angle and half-angle formulas

One often needs to deal with expressions in which two or more angles are involved, such as \displaystyle \sin(u+v). One will then need the so-called " addition formulas “. For sine and cosine the formulas are

If one wants to know the sine or cosine of a double angle, that is \displaystyle \sin 2v or \displaystyle \cos 2v, one can write these expressions as \displaystyle \sin(v + v) or \displaystyle \cos(v + v) and use the addition formulas above and get the double-angle

From these relationships, one can then get the formulas for half angles. By replacing \displaystyle 2v by \displaystyle v, and consequently \displaystyle v by \displaystyle v/2, in the formula for \displaystyle \cos 2v one gets that Vorlage:Fristående formel

If we want a formula for \displaystyle \sin(v/2) we use the Pythagorean identity to get rid of \displaystyle \cos^2(v/2) Vorlage:Fristående formel i.e.

Similarly, we can use the Pythagorean identity to get rid of \displaystyle \sin^2(v/2). Then we will have instead

Exercises