4. Trigonometry
From Förberedande kurs i matematik 1
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| - | ''' | + | '''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 about the origins of geometry and trigonometry.'' |
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| - | === | + | ===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. | |
| - | + | 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 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! | ||
| - | 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. We will also see how 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''' | ||
| - | + | 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 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 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. | |
| - | 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. | ||
| + | [[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. | ||
| - | + | 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. | |
| - | + | ||
| - | 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. | ||
| - | + | A knowledge of geometry is also very useful in everyday life, where one is often faced with questions of a geometrical nature. | |
| - | ''' | + | '''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"> | + | <div class="inforuta" style="width:580px;"> |
| - | ''' | + | '''To become skilled in Trigonometry''' |
| - | # | + | # 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. |
| - | # | + | # 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. |
| - | # | + | # 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. | + | 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> | ||
Current revision
| \displaystyle \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)} |
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 about the origins of geometry and trigonometry.
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.
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 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!
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.
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.
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 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.
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 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.
A knowledge of geometry is also very useful in everyday life, where one is often faced with questions of a geometrical nature.
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.
To become skilled in Trigonometry
- 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.
- 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.
- 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 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.
