4. Trigonometry
From Förberedande kurs i matematik 1
m |
|||
| (9 intermediate revisions not shown.) | |||
| Line 1: | Line 1: | ||
__NOTOC__ | __NOTOC__ | ||
| - | <!-- Don't remove this line --> | ||
| + | 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 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! | |
| - | + | ||
| - | + | ||
| - | + | ||
| - | '''How old are geometry and trigonometry and when did one start to use these methods to solve problems? ''' | ||
| - | + | 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. | |
| - | + | ||
| - | + | ||
| - | + | ||
| - | + | ||
| - | + | ||
| - | + | ||
| - | + | ||
| - | + | ||
| - | + | ||
| - | + | ||
| - | + | ||
| - | + | ||
| - | + | ||
| - | + | ||
| - | + | ||
| - | + | ||
| - | In this chapter we will see some examples of how geometric objects such as lines, parabolas and circles are described by equations. | + | |
| Line 34: | Line 14: | ||
'''The unit circle is of particular importance''' | '''The unit circle is of particular importance''' | ||
| - | The circle | + | 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, | + | 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 | + | The functions cosine and sine are thus used to relate angles to distances. |
| - | + | 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]] |
| - | + | 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 | + | 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. |
| Line 52: | Line 32: | ||
| - | '''It is important to note that the material in this section— as well as in other parts of the course — is designed that | + | '''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 | + | '''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 | + | # 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. | # 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 | + | # 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 | + | 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
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 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!
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 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, 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.
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 \displaystyle 0 and \displaystyle \pi/2 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.
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.
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.
