4. Trigonometry - Revision history

Tek at 13:34, 29 January 2009

2009-01-29T13:34:58Z

←Older revision		Revision as of 13:34, 29 January 2009
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-	\| 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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-	'''When did people first start studying geometry? When did people start using trigonometry to solve geometrical problems?'''
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-	''Watch the video in which the lecturer Lasse Svensson tells us about the origins of geometry and trigonometry.''
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-	===What is geometry? ===
	Geometry is a very old science. The word comes from ancient Greek, and means, literally, "earth-measuring"; it has come to have the broader meaning ”the science of space”. The study of shape and space is truly ancient; even in its systematic, Greek form, it began several centuries BC.		Geometry is a very old science. The word comes from ancient Greek, and means, literally, "earth-measuring"; it has come to have the broader meaning ”the science of space”. The study of shape and space is truly ancient; even in its systematic, Greek form, it began several centuries BC.

Phil at 16:52, 6 January 2009

2009-01-06T16:52:31Z

←Older revision		Revision as of 16:52, 6 January 2009
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	Geometry is a very old science. The word comes from ancient Greek, and means, literally, "earth-measuring"; it has come to have the broader meaning ”the science of space”. The study of shape and space is truly ancient; even in its systematic, Greek form, it began several centuries BC.		Geometry is a very old science. The word comes from ancient Greek, and means, literally, "earth-measuring"; it has come to have the broader meaning ”the science of space”. The study of shape and space is truly ancient; even in its systematic, Greek form, it began several centuries BC.

-	Perhaps the most famous of the ~~eraly~~ Greek geometers 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 (it underlies our modern understanding of gravity, for example).	+	Perhaps the most famous of the early Greek geometers 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 (it underlies our modern understanding of gravity, for example).

	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 BC. One of the most famous mathematicians who developed the theory was HIPPARCHUS, who studied circles and ''chords'' (a chord is a line segment joining two points on the circumference). 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 BC. One of the most famous mathematicians who developed the theory was HIPPARCHUS, who studied circles and ''chords'' (a chord is a line segment joining two points on the circumference). 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!

Phil: Edited for UK idiom and terminology; small change to the historical emphasis pending broader discussion of this issue.

2009-01-05T17:45:13Z

Edited for UK idiom and terminology; small change to the historical emphasis pending broader discussion of this issue.

Jdinglis at 17:11, 10 November 2008

2008-11-10T17:11:21Z

←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

2008-11-10T14:59:31Z

Jdinglis at 14:42, 10 November 2008

2008-11-10T14:42:20Z

←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

2008-09-10T08:37:14Z

←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

2008-09-09T12:20:23Z

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:)

2008-08-21T06:19:29Z

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

2008-08-19T12:53:19Z

small fixes

←Older revision		Revision as of 12:53, 19 August 2008
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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.
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←Older revision		Revision as of 17:45, 5 January 2009
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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 studied geometry~~.	+	Geometry is a very old science. The word comes from ancient Greek, and means, literally, "earth-measuring"; it has come to have the broader meaning ”the science of space”. The study of shape and space is truly ancient; even in its systematic, Greek form, it began several centuries BC.

-	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 the eraly Greek geometers 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 (it underlies our modern understanding of gravity, for example).

-	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!	+	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 BC. One of the most famous mathematicians who developed the theory was HIPPARCHUS, who studied circles and ''chords'' (a chord is a line segment joining two points on the circumference). 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 the trigonometric functions cosine and sine.	+	The circle whose radius is 1 unit and whose centre is the origin is especially important. One can use this circle to introduce various concepts regarding angles as well as the trigonometric functions cosine and sine.

-	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~~.	+	An angle corresponds to a point on the unit circle. The angle, measured in ''radians'', can be thought of as simply the distance along the circle from this point to the point (1,0). The cosine of the angle is just the ''x''-coordinate of the point and the sine of the angle its ''y''-coordinate.

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

-	~~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.	+	You may well be accustomed to think of cosine and sine as relations between the sides of a right-angled triangle. This approach is all right as far as it goes, but it only works for angles between 0 and 90 degrees (that is, between <math>0</math> and <math>\pi/2</math> radians). For more general angles, 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]]
-	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.	+	Managing and manipulating trigonometric expressions is an important skill, used in lots of applications of mathematics. Thus the final section provides an exercise in which you can practise these skills thoroughly.

	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.		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.

←Older revision		Revision as of 14:59, 10 November 2008
Line 21:		Line 21:

	===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.'''



	<div class="inforuta" style="width:580px;">		<div class="inforuta" style="width:580px;">
-	'''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.

	</div>		</div>