4. Trigonometry - Revision history

Jdinglis at 17:11, 10 November 2008

Jdinglis — Mon, 10 Nov 2008 17:11:21 GMT

←Older revision		Revision as of 17:11, 10 November 2008
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	Perhaps the most famous of these is '''EUCLID''' (who lived around 300 BC). He wrote a famous work entitled '''ELEMENTS''', in which he summed up the mathematical knowledge of his time. In the 17th century people began to call into question the validity of some of the so-called Euclidean '''AXIOMS''' and a '''NON-EUCLIDEAN''' geometry was developed which became of great importance in different contexts.		Perhaps the most famous of these is '''EUCLID''' (who lived around 300 BC). He wrote a famous work entitled '''ELEMENTS''', in which he summed up the mathematical knowledge of his time. In the 17th century people began to call into question the validity of some of the so-called Euclidean '''AXIOMS''' and a '''NON-EUCLIDEAN''' geometry was developed which became of great importance in different contexts.

-	Trigonometry comes from Greek ("trigonon" stands for "triangle" and "metron" stands for "measure") and is a method to calculate the angles and sides of right-angled triangles. Trigonometry was developed a few hundred years before the birth of Christ. One of the most famous mathematicians was HIPPARCHUS, who studied circles and chords. For each chord, he was able to calculate the corresponding arc length and in this way, he was able to determine the sides and angles of triangles. All this took place 2200 years before the advent of the calculator!	+	Trigonometry comes from Greek ("trigonon" stands for "triangle" and "metron" stands for "measure") and is a method to calculate the angles and sides of right-angled triangles. Trigonometry was developed a few hundred years before the birth of Christ. One of the most famous mathematicians who developed the theory was HIPPARCHUS, who studied circles and chords. For each chord, he was able to calculate the corresponding arc length and in this way, he was able to determine the sides and angles of triangles. All this took place 2200 years before the advent of the calculator!

Jdinglis at 14:59, 10 November 2008

Jdinglis — Mon, 10 Nov 2008 14:59:31 GMT

Jdinglis at 14:42, 10 November 2008

Jdinglis — Mon, 10 Nov 2008 14:42:20 GMT

←Older revision		Revision as of 14:42, 10 November 2008
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-	'''~~How old are~~ geometry ~~and trigonometry~~ ~~and when~~ did ~~one~~ start ~~to use these methods~~ to solve problems? '''	+	'''When did people first start studying geometry? When did people start using trigonometry to solve geometrical problems?'''


-	''Watch the video in which the lecturer Lasse Svensson tells us ~~how~~ geometry and trigonometry ~~developed and answers Elins questions~~.''	+	''Watch the video in which the lecturer Lasse Svensson tells us about the origins of geometry and trigonometry.''

Tek at 08:37, 10 September 2008

Tek — Wed, 10 Sep 2008 08:37:14 GMT

←Older revision		Revision as of 08:37, 10 September 2008
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-	<div class="inforuta">	+	<div class="inforuta" style="width:580px;">
	'''To do well in Trigonometry'''		'''To do well in Trigonometry'''

Tekbot: 4. Trigonometri moved to 4. Trigonometry: Robot: moved page

Tekbot — Tue, 09 Sep 2008 12:20:23 GMT

4. Trigonometri moved to 4. Trigonometry: Robot: moved page

←Older revision

Revision as of 12:20, 9 September 2008

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

Tekbot — Thu, 21 Aug 2008 06:19:29 GMT

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

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-	[[~~Bild~~:cikel.jpg\|right]]	+	[[Image:cikel.jpg\|right]]
	To be able to manage and manipulate trigonometric expressions is important in most applications of mathematics. Thus the final section provides a thorough exercise to practise these skills.		To be able to manage and manipulate trigonometric expressions is important in most applications of mathematics. Thus the final section provides a thorough exercise to practise these skills.

Tek: small fixes

Tek — Tue, 19 Aug 2008 12:53:19 GMT

small fixes

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	# Start by reading the section's theory and study the examples.		# Start by reading the section's theory and study the examples.
-	# Work through the ~~practice questions~~ and try to solve them without using a calculator. Make sure that you have the right answer by clicking on the answer button. If you do not have it, you can click on the solution button to see what went wrong	+	# Work through the exercises and try to solve them without using a calculator. Make sure that you have the right answer by clicking on the answer button. If you do not have it, you can click on the solution button to see what went wrong
	# Then go ahead and answer the questions in the basic test of the section.		# Then go ahead and answer the questions in the basic test of the section.
	# If you get stuck on a point, check to see if someone else has discussed the point in the forum belonging to the section. If not, take up the point yourself. Your teacher (or a student) will respond to your question within a few hours.		# If you get stuck on a point, check to see if someone else has discussed the point in the forum belonging to the section. If not, take up the point yourself. Your teacher (or a student) will respond to your question within a few hours.
	# When you have answered correctly all questions in both the basic and the final test of this section you will have a pass for this section and then you should move on to Part 5 and work with an individual assignment and group assignment. Links to these are to be found in the "Student Lounge."		# When you have answered correctly all questions in both the basic and the final test of this section you will have a pass for this section and then you should move on to Part 5 and work with an individual assignment and group assignment. Links to these are to be found in the "Student Lounge."

-	PS. If you feel that you are very familiar with the contents of a section you can test yourself by going directly to the basic and final tests. You must answer all the questions correctly in a test, but you may do the test several times if you do not succeed at the first attempt. It is your final results which appear in the statistics.	+	PS. If you feel that you are very familiar with the contents of a section you can test yourself by going directly to the basic and final tests. You must answer all the questions correctly in a test, but you may do the test several times if you do not succeed at the first attempt. It is your final results which appear in the statistics.
-	\|}	+

	</div>		</div>

Ian at 17:10, 17 July 2008

Ian — Thu, 17 Jul 2008 17:10:58 GMT

←Older revision		Revision as of 17:10, 17 July 2008
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	Perhaps the most famous of these is '''EUCLID''' (around 300 BCE). He wrote a famous work entitled '''ELEMENTS''' - in which he summed up the mathematical knowledge of his time. In the 17th century one began to call into question some of the so-called Euclid '''AXIOMS''' and a '''NON-EUCLIDEAN''' geometry was developed which became of great importance in different contexts.		Perhaps the most famous of these is '''EUCLID''' (around 300 BCE). He wrote a famous work entitled '''ELEMENTS''' - in which he summed up the mathematical knowledge of his time. In the 17th century one began to call into question some of the so-called Euclid '''AXIOMS''' and a '''NON-EUCLIDEAN''' geometry was developed which became of great importance in different contexts.

-	Trigonometry comes from Greek ("trigonon" stands for "triangle" and "metron" stands for "measure") and is a method to calculate the angles and sides of right-angled triangles. Trigonometry developed a few hundred years before the birth of Christ. One of the most famous mathematicians ~~were~~ HIPPARCHUS, who worked with the circle and chords within a circle. For each chord, he was able to calculate the corresponding arc length and in this way, he was able to determine the sides and angles of triangles. All this took place 2200 years before the advent of the ~~calculators~~!	+	Trigonometry comes from Greek ("trigonon" stands for "triangle" and "metron" stands for "measure") and is a method to calculate the angles and sides of right-angled triangles. Trigonometry developed a few hundred years before the birth of Christ. One of the most famous mathematicians was HIPPARCHUS, who worked with the circle and chords within a circle. For each chord, he was able to calculate the corresponding arc length and in this way, he was able to determine the sides and angles of triangles. All this took place 2200 years before the advent of the calculator!


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	'''The unit circle is of particular importance'''		'''The unit circle is of particular importance'''

-	The circle with a radius of 1 around the origin is especially important. One can use this circle to introduce the various concepts regarding angles as well as ~~and~~ the trigonometric functions cosine and sine.	+	The circle with a radius of 1 around the origin is especially important. One can use this circle to introduce the various concepts regarding angles as well as the trigonometric functions cosine and sine.

-	An angle corresponds to a point on the circle, the ~~unit~~ angle is the distance along the circle to the point (1.0), the cosine of the angle is the "x"-component of the point, the sine of the angle is "y"-component of the point.	+	An angle corresponds to a point on the unit circle, the measure of the angle is the distance along the circle to the point (1.0), the cosine of the angle is the "x"-component of the point, the sine of the angle is "y"-component of the point.

	The functions cosine and sine thus are used to translate from angles to distances.		The functions cosine and sine thus are used to translate from angles to distances.

Ian at 08:00, 15 July 2008

Ian — Tue, 15 Jul 2008 08:00:19 GMT

←Older revision		Revision as of 08:00, 15 July 2008
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-	'''~~Hur gammal är geometrin och trigonometrin egentligen och när började man använda dessa metoder för att lösa problem~~?'''	+	'''How old are geometry and trigonometry and when did one start to use these methods to solve problems? '''


-	''~~Titta på videon där universitetslektor~~ Lasse Svensson ~~berättar om hur geometri och trigonometri utvecklats och svarar på~~ Elins ~~frågor~~.''	+	''Watch the video in which the lecturer Lasse Svensson tells us how geometry and trigonometry developed and answers Elins questions.''


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-	===~~Vad är geometri~~?===	+	===What is geometry? ===
-	~~Geometri är en mycket gammal vetenskap~~. ~~Geometri är grekiska och betyder ”läran om rummet”~~. "Ge" ~~står för jord och~~ "~~metrein~~" ~~betyder mäta~~. ~~Långt före Jesu födelse hade skickliga matematiker utvecklat geometrin~~.	+	Geometry is a very old science. Geometry comes from Greek and means ”science of space”. "Ge" stands for earth and "metry" for science of measuring. Long before the birth of Jesus brilliant mathematicians had developed geometry.

-	~~Den kanske mest berömde är~~ '''~~EUKLIDES~~''' (300~~-talet f.Kr~~). ~~Han gav ut ett berömt verk med titeln~~ '''~~ELEMENTA~~''' - ~~där han sammanfattade sin tids matematiska vetande~~. ~~På 1600~~-~~talet började man ifrågasätta en del av Euklides s.k.~~ '''~~AXIOM~~''' ~~och det utvecklades en~~ '''~~ICKE~~-~~EUKLIDISK~~''' ~~geometri som fick stor betydelse i olika sammanhang~~.	+	Perhaps the most famous of these is '''EUCLID''' (around 300 BCE). He wrote a famous work entitled '''ELEMENTS''' - in which he summed up the mathematical knowledge of his time. In the 17th century one began to call into question some of the so-called Euclid '''AXIOMS''' and a '''NON-EUCLIDEAN''' geometry was developed which became of great importance in different contexts.

		+	Trigonometry comes from Greek ("trigonon" stands for "triangle" and "metron" stands for "measure") and is a method to calculate the angles and sides of right-angled triangles. Trigonometry developed a few hundred years before the birth of Christ. One of the most famous mathematicians were HIPPARCHUS, who worked with the circle and chords within a circle. For each chord, he was able to calculate the corresponding arc length and in this way, he was able to determine the sides and angles of triangles. All this took place 2200 years before the advent of the calculators!

-	Trigonometri betyder "triangelmätning" och är en metod för att beräkna vinklar och sidor i rätvinkliga trianglar. Trigonometrin utvecklades några hundra år före Kristi födelse. En av de mest kända matematikerna då var HIPPARKUS, som arbetade med cirkeln och kordor i cirkeln. För varje korda kunde han beräkna motsvarande cirkelbåges längd och på så sätt kunde han bestämma sidor och vinklar i trianglar. Detta hände 2200 år före miniräknarens tillkomst!

		+	In this chapter we will see some examples of how geometric objects such as lines, parabolas and circles are described by equations. Similarly various regions can be described by inequalities.

-	I detta kapitel ska vi se några exempel på hur geometriska objekt som linjer, parabler och cirklar beskrivs av ekvationer. På liknande sätt kan olika områden beskrivas av olikheter.


		+	'''The unit circle is of particular importance'''

-	~~'''Enhetscirkeln är särskilt viktig'''~~	+	The circle with a radius of 1 around the origin is especially important. One can use this circle to introduce the various concepts regarding angles as well as and the trigonometric functions cosine and sine.

-	~~Cirkeln med radie~~ 1 ~~runt origo är speciellt viktig~~. ~~I denna cirkel kan man sammanfatta olika vinkelbegrepp samt de trigonometriska funktionerna cosinus och sinus~~.	+	An angle corresponds to a point on the circle, the unit angle is the distance along the circle to the point (1.0), the cosine of the angle is the "x"-component of the point, the sine of the angle is "y"-component of the point.

-	En vinkel svarar mot en punkt på enhetscirkeln, dess vinkelmått är sträckan längs cirkeln till punkten (1,0), cosinus för vinkeln är punktens ''x''-komponent, sinus för vinkeln är ''y''-komponenten av punkten.	+	The functions cosine and sine thus are used to translate from angles to distances.

-	~~Funktionerna cosinus och sinus används alltså för att översätta från vinklar till sträckor~~.	+	If one is accustomed to think of cosine and sine as relations between the sides of a right-angled triangle, it is extremely important to rethink these functions in terms of the unit circle. This way it will be easier to understand trigonometric relationships like periodicity and the Pythagorean identity, the relationships for doubling angles and the formulas for derivatives.
-		+
-	Om man är van att tänka på cosinus och sinus som förhållanden mellan sidorna i en rätvinklig triangel så är det viktigt att också fundera ordentligt på dessa funktioner i enhetscirkeln. Denna bild gör det lättare att förstå trigonometriska samband som periodicitet, trigonometriska ettan, ~~samband för dubblering av vinkeln samt formler för derivator~~.	+


	[[Bild:cikel.jpg\|right]]		[[Bild:cikel.jpg\|right]]
-	~~Att hantera och manipulera med trigonometriska uttryck är viktigt inom de flesta tillämpningar av matematiken~~. ~~De avslutande avsnitten ger en grundlig övning på detta~~.	+	To be able to manage and manipulate trigonometric expressions is important in most applications of mathematics. Thus the final section provides a thorough exercise to practise these skills.
		+
		+	Once geometry was one of the main elements in a mathematics course. In recent decades, classical geometry has decreased both in high school as well as in university's courses. But, for anyone who intends to be active in photography or graphics or with construction and design (such as CAD), a good knowledge of geometry is very valuable.

-	Tidigare räknades geometrin till de viktigare momenten inom matematikundervisningen. Under de senaste decennierna har den klassiska geometrin minskats ner i såväl gymnasiets som i högskolans kurser. Men, för den som kommer att syssla med bilder eller grafik eller med konstruktioner och design (t.ex. CAD), så är goda kunskaper i geometri mycket värdefulla.

-	~~Kunskaper i geometri är också väldigt bra att ha med sig ut i vardagslivet~~, ~~där man ofta ställs inför geometriska problem och funderingar~~.	+	A knowledge of geometry is also very useful in everyday life, where one is often faced with questions of a geometrical nature.


-	'''~~Observera att materialet i denna kursdel – liksom i övriga delar av kursen – är utformat för att man ska arbeta med det utan hjälp av miniräknare~~.'''	+	'''It is important to note that the material in this section— as well as in other parts of the course — is designed that one does not use calculators.'''



	<div class="inforuta">		<div class="inforuta">
-	'''~~Så här lyckas du med Trigonometri~~'''	+	'''To do well in Trigonometry'''

-	# ~~Börja med att läsa genomgången till ett avsnitt och tänka igenom exemplen~~.	+	# Start by reading the section's theory and study the examples.
-	# ~~Arbeta sedan med övningsuppgifterna och försök att lösa dem utan miniräknare~~. ~~Kontrollera att du kommit fram till rätt svar genom att klicka på svarsknappen~~. ~~Har du inte det~~, ~~så kan du klicka på lösningsknappen, för att se hur du ska göra.~~	+	# Work through the practice questions and try to solve them without using a calculator. Make sure that you have the right answer by clicking on the answer button. If you do not have it, you can click on the solution button to see what went wrong
-	# ~~Gå därefter vidare och svara på frågorna i grundprovet som hör till avsnittet~~.	+	# Then go ahead and answer the questions in the basic test of the section.
-	# ~~Skulle du fastna~~, ~~se efter om någon ställt en fråga om just detta i avsnittets~~ forum. ~~Ställ annars en fråga om du undrar över något~~. ~~Din lärare~~ (~~eller en studiekamrat~~) ~~kommer att besvara den inom några timmar~~.	+	# If you get stuck on a point, check to see if someone else has discussed the point in the forum belonging to the section. If not, take up the point yourself. Your teacher (or a student) will respond to your question within a few hours.
-	# När du är klar med övningsuppgifterna och grundproven i ett avsnitt så ska du göra slutprovet för att bli godkänd på avsnittet. Där gäller det att svara rätt på tre frågor i följd för att kunna gå vidare.	+	# When you have answered correctly all questions in both the basic and the final test of this section you will have a pass for this section and then you should move on to Part 5 and work with an individual assignment and group assignment. Links to these are to be found in the "Student Lounge."
-	~~# När du fått alla rätt på både grundprov och slutprov, så ska du gå vidare till Del~~ 5 ~~och arbeta med en inlämnings- och gruppuppgiften~~. ~~Länk till dessa hittar du i~~ "Student Lounge".	+

-	PS. ~~Tycker du att innehållet i ett avsnitt känns väldigt bekant, så kan du testa att gå direkt till grundprovet och slutprovet~~. ~~Du måste få alla rätt på ett prov, men kan göra om provet flera gånger~~, ~~om du inte lyckas på första försöket~~. ~~Det är ditt senaste resultat som visas i statistiken~~.	+	PS. If you feel that you are very familiar with the contents of a section you can test yourself by going directly to the basic and final tests. You must answer all the questions correctly in a test, but you may do the test several times if you do not succeed at the first attempt. It is your final results which appear in the statistics.
	\|}		\|}

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Tek — Mon, 05 May 2008 14:30:19 GMT

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{|align="left"
| width="220" height="203" |<math>\text{@(a class="image" href="http://smaug.nti.se/temp/KTH/film5.html" target="_blank")@(img src="http://wiki.math.se/wikis/2008/forberedandematte1/img_auth.php/0/00/Lars_och_Elin.jpg" alt="Film om trigonometri")@(/img)@(/a)}</math>
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'''Hur gammal är geometrin och trigonometrin egentligen och när började man använda dessa metoder för att lösa problem?'''

''Titta på videon där universitetslektor Lasse Svensson berättar om hur geometri och trigonometri utvecklats och svarar på Elins frågor.''

===Vad är geometri?===
Geometri är en mycket gammal vetenskap. Geometri är grekiska och betyder ”läran om rummet”. "Ge" står för jord och "metrein" betyder mäta. Långt före Jesu födelse hade skickliga matematiker utvecklat geometrin.

Den kanske mest berömde är '''EUKLIDES''' (300-talet f.Kr). Han gav ut ett berömt verk med titeln '''ELEMENTA''' - där han sammanfattade sin tids matematiska vetande. På 1600-talet började man ifrågasätta en del av Euklides s.k. '''AXIOM''' och det utvecklades en '''ICKE-EUKLIDISK''' geometri som fick stor betydelse i olika sammanhang.

Trigonometri betyder "triangelmätning" och är en metod för att beräkna vinklar och sidor i rätvinkliga trianglar. Trigonometrin utvecklades några hundra år före Kristi födelse. En av de mest kända matematikerna då var HIPPARKUS, som arbetade med cirkeln och kordor i cirkeln. För varje korda kunde han beräkna motsvarande cirkelbåges längd och på så sätt kunde han bestämma sidor och vinklar i trianglar. Detta hände 2200 år före miniräknarens tillkomst!

I detta kapitel ska vi se några exempel på hur geometriska objekt som linjer, parabler och cirklar beskrivs av ekvationer. På liknande sätt kan olika områden beskrivas av olikheter.

'''Enhetscirkeln är särskilt viktig'''

Cirkeln med radie 1 runt origo är speciellt viktig. I denna cirkel kan man sammanfatta olika vinkelbegrepp samt de trigonometriska funktionerna cosinus och sinus.

En vinkel svarar mot en punkt på enhetscirkeln, dess vinkelmått är sträckan längs cirkeln till punkten (1,0), cosinus för vinkeln är punktens ''x''-komponent, sinus för vinkeln är ''y''-komponenten av punkten.

Funktionerna cosinus och sinus används alltså för att översätta från vinklar till sträckor.

Om man är van att tänka på cosinus och sinus som förhållanden mellan sidorna i en rätvinklig triangel så är det viktigt att också fundera ordentligt på dessa funktioner i enhetscirkeln. Denna bild gör det lättare att förstå trigonometriska samband som periodicitet, trigonometriska ettan, samband för dubblering av vinkeln samt formler för derivator.

[[Bild:cikel.jpg|right]]
Att hantera och manipulera med trigonometriska uttryck är viktigt inom de flesta tillämpningar av matematiken. De avslutande avsnitten ger en grundlig övning på detta.

Tidigare räknades geometrin till de viktigare momenten inom matematikundervisningen. Under de senaste decennierna har den klassiska geometrin minskats ner i såväl gymnasiets som i högskolans kurser. Men, för den som kommer att syssla med bilder eller grafik eller med konstruktioner och design (t.ex. CAD), så är goda kunskaper i geometri mycket värdefulla.

Kunskaper i geometri är också väldigt bra att ha med sig ut i vardagslivet, där man ofta ställs inför geometriska problem och funderingar.

'''Observera att materialet i denna kursdel – liksom i övriga delar av kursen – är utformat för att man ska arbeta med det utan hjälp av miniräknare.'''

<div class="inforuta">
'''Så här lyckas du med Trigonometri'''

# Börja med att läsa genomgången till ett avsnitt och tänka igenom exemplen.
# Arbeta sedan med övningsuppgifterna och försök att lösa dem utan miniräknare. Kontrollera att du kommit fram till rätt svar genom att klicka på svarsknappen. Har du inte det, så kan du klicka på lösningsknappen, för att se hur du ska göra.
# Gå därefter vidare och svara på frågorna i grundprovet som hör till avsnittet.
# Skulle du fastna, se efter om någon ställt en fråga om just detta i avsnittets forum. Ställ annars en fråga om du undrar över något. Din lärare (eller en studiekamrat) kommer att besvara den inom några timmar.
# När du är klar med övningsuppgifterna och grundproven i ett avsnitt så ska du göra slutprovet för att bli godkänd på avsnittet. Där gäller det att svara rätt på tre frågor i följd för att kunna gå vidare.
# När du fått alla rätt på både grundprov och slutprov, så ska du gå vidare till Del 5 och arbeta med en inlämnings- och gruppuppgiften. Länk till dessa hittar du i "Student Lounge".

   PS. Tycker du att innehållet i ett avsnitt känns väldigt bekant, så kan du testa att gå direkt till grundprovet och slutprovet. Du måste få alla rätt på ett prov, men kan göra om provet flera gånger, om du inte lyckas på första försöket. Det är ditt senaste resultat som visas i statistiken.
|}

</div>

←Older revision		Revision as of 14:59, 10 November 2008
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	===What is geometry? ===		===What is geometry? ===
-	Geometry is a very old science. Geometry comes from Greek and means ”science of space”. "Ge" stands for earth and "metry" for science of measuring. Long before the birth of Jesus brilliant mathematicians had ~~developed~~ geometry.	+	Geometry is a very old science. Geometry comes from Greek and means ”science of space”. "Ge" stands for earth and "metry" for science of measuring. Long before the birth of Jesus brilliant mathematicians had studied geometry.

-	Perhaps the most famous of these is '''EUCLID''' (around 300 ~~BCE~~). He wrote a famous work entitled '''ELEMENTS''' - in which he summed up the mathematical knowledge of his time. In the 17th century ~~one~~ began to call into question some of the so-called ~~Euclid~~ '''AXIOMS''' and a '''NON-EUCLIDEAN''' geometry was developed which became of great importance in different contexts.	+	Perhaps the most famous of these is '''EUCLID''' (who lived around 300 BC). He wrote a famous work entitled '''ELEMENTS''', in which he summed up the mathematical knowledge of his time. In the 17th century people began to call into question the validity of some of the so-called Euclidean '''AXIOMS''' and a '''NON-EUCLIDEAN''' geometry was developed which became of great importance in different contexts.

-	Trigonometry comes from Greek ("trigonon" stands for "triangle" and "metron" stands for "measure") and is a method to calculate the angles and sides of right-angled triangles. Trigonometry developed a few hundred years before the birth of Christ. One of the most famous mathematicians was HIPPARCHUS, who ~~worked with the circle~~ and chords ~~within a circle~~. For each chord, he was able to calculate the corresponding arc length and in this way, he was able to determine the sides and angles of triangles. All this took place 2200 years before the advent of the calculator!	+	Trigonometry comes from Greek ("trigonon" stands for "triangle" and "metron" stands for "measure") and is a method to calculate the angles and sides of right-angled triangles. Trigonometry was developed a few hundred years before the birth of Christ. One of the most famous mathematicians was HIPPARCHUS, who studied circles and chords. For each chord, he was able to calculate the corresponding arc length and in this way, he was able to determine the sides and angles of triangles. All this took place 2200 years before the advent of the calculator!


-	In this chapter we will see some examples of how geometric objects such as lines, parabolas and circles are described by equations. ~~Similarly~~ various regions can be described by inequalities.	+	In this chapter we will see some examples of how geometric objects such as lines, parabolas and circles are described by equations. We will also see how various regions can be described by inequalities.


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	The circle with a radius of 1 around the origin is especially important. One can use this circle to introduce the various concepts regarding angles as well as the trigonometric functions cosine and sine.		The circle with a radius of 1 around the origin is especially important. One can use this circle to introduce the various concepts regarding angles as well as the trigonometric functions cosine and sine.

-	An angle corresponds to a point on the unit circle~~, the measure of the~~ angle is the distance along the circle to the point (1.0)~~, the~~ cosine of the angle is the "x"-component of the point, the sine of the angle is "y"-component of the point.	+	An angle corresponds to a point on the unit circle. The angle is measured by the distance along the circle from the point to the point (1,0). The cosine of the angle is the "x"-component of the point and the sine of the angle is the "y"-component of the point.

-	The functions cosine and sine ~~thus~~ are used to ~~translate from~~ angles to distances.	+	The functions cosine and sine are thus used to relate angles to distances.

-	If ~~one is~~ accustomed to think of cosine and sine as relations between the sides of a right-angled triangle, it is extremely important to rethink these functions in terms of the unit circle. This way it will be easier to understand trigonometric relationships like periodicity ~~and~~ the Pythagorean identity, the ~~relationships for doubling angles~~ and the formulas for derivatives.	+	If you are accustomed to think of cosine and sine as relations between the sides of a right-angled triangle, it is extremely important to rethink these functions in terms of the unit circle. This way it will be easier to understand trigonometric relationships like periodicity, the Pythagorean identity, the double angle formulas and the formulas for the derivatives of trigonometric functions.


	[[Image:cikel.jpg\|right]]		[[Image:cikel.jpg\|right]]
-	~~To be able to manage~~ and ~~manipulate~~ trigonometric expressions is important in ~~most~~ applications of mathematics. Thus the final section provides a thorough exercise to practise these skills.	+	Managing and manipulating trigonometric expressions is an important skill, used in lots of applications of mathematics. Thus the final section provides a thorough exercise in which you can practise these skills.

-	Once geometry was one of the main elements in a mathematics course. In recent decades, classical geometry ~~has decreased~~ both in high school ~~as well as in~~ university's courses. ~~But~~, for anyone who intends to be active in photography or graphics or with construction and design (such as CAD), a good knowledge of geometry is very valuable.	+	Once geometry was one of the main elements in a mathematics course. In recent decades the amount of classical geometry taught in both high school and university courses has decreased. However, for anyone who intends to be active in photography or graphics or with construction and design (such as CAD), a good knowledge of geometry is very valuable.


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-	'''It is important to note that the material in this section— as well as in other parts of the course — is designed that ~~one does not~~ use ~~calculators~~.'''	+	'''It is important to note that the material in this section— as well as in other parts of the course — is designed so that you don't have to use a calculator.'''



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-	'''To ~~do well~~ in Trigonometry'''	+	'''To become skilled in Trigonometry'''

	# Start by reading the section's theory and study the examples.		# Start by reading the section's theory and study the examples.
-	# Work through the exercises and try to solve them without using a calculator. Make sure that you have the right answer by clicking on the answer button. If you do not have it, you can click on the solution button to see what went wrong	+	# Work through the exercises and try to solve them without using a calculator. Make sure that you have the right answer by clicking on the answer button. If you do not have it, you can click on the solution button to see what went wrong.
-	# ~~Then go ahead and answer~~ the questions in the basic test of the section.	+	# Answer the questions in the basic test of the section.
	# If you get stuck on a point, check to see if someone else has discussed the point in the forum belonging to the section. If not, take up the point yourself. Your teacher (or a student) will respond to your question within a few hours.		# If you get stuck on a point, check to see if someone else has discussed the point in the forum belonging to the section. If not, take up the point yourself. Your teacher (or a student) will respond to your question within a few hours.
-	# When you have answered correctly all questions in both the basic and the final test of this section you will have a pass for this section ~~and then~~ you should move on to Part 5 and work with an individual assignment and group assignment. Links to these are to be found in the "Student Lounge."	+	# When you have answered correctly all questions in both the basic and the final test of this section you will have a pass for this section. Then you should move on to Part 5 and work with an individual assignment and group assignment. Links to these are to be found in the "Student Lounge."

-	PS. If you feel that you are very familiar with the contents of a section you can test yourself by going directly to the ~~basic and final~~ tests. You must answer all the questions correctly in a test, but you may do the test several times if you do not succeed at the first attempt. It is your final results which appear in the statistics.	+	PS. If you feel that you are very familiar with the contents of a section you can test yourself by going directly to the tests. You must answer all the questions correctly in a test, but you may do the test several times if you do not succeed at the first attempt. It is your final results which appear in the statistics.

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