From Förberedande kurs i matematik 1

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.