As this picture of a sliced poultry sausage shows — and as we see it every day “*at the butcher’s*” — sausages are very often cut at an angle. Geometry teaches us that the intersection is then not bounded by a *circle* but by an *ellipse*. If one cuts the sausage shell parallel to the “*main axis*” of the sausage and places it on a flat surface, the initially “*spatial cut*” becomes a flat curve that represents part of a so-called *sine curve*, i.e. a *harmonic oscillation*.

The associated exhibit in the Maths Adventure Land (see Figure 2 below) shows how a hand crank is used to map the plane, oblique section of a straight (circular) *cylinder* onto an endless sheet. The image on the slide turns out to be a harmonic oscillation. It is mathematically described by an angular function (on the right-angled) triangle, the so-called *sine*.

### And now … the mathematics:

#### 1. definition of the sine function

According to the following figure 3, the sine (also: *sine function*) of an angle is the length of the so-called *opposite cathetus* in a *right triangle* with a *hypotenuse* of length one.

By means of a *unit circle* (*radius* ) one assigns to each angle the length of the arc over this angle . Considering that the *circumference* of the unit circle is equal to , the corresponding radian is :

Table 1: Relationship between degrees and radians

The sine function is defined by . The length of the so-called *adjacent cathetus* in a right triangle with a *hypotenuse* of length 1 over the angle with the *arc length* is called the *cosine* (also: *cosine function*) with .

#### 2. unwinding of the plane cut

A (straight) *circular cylinder* with radius for the *base circle* (in the ADVENTURELAND MATHEMATICS ) is described by the following equations because of the equation (*Pythagorean theorem* on the unit circle):

Here and are the so-called *polar coordinates*, as shown in the following figure 4:

A plane section of the circular cylinder (at the angle of ) is given by the *bisector* in the -plane

From equations (1) and (2) thus follows

for . This sine function is visible with on the slide to be unrolled at the corresponding exhibit in the ADVENTURELAND MATHEMATICS.