The “Sine”

Figure 1: Poultry sausage. Where is there a sine here?

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

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

Figure 2: Exhibit in the Mathematics Adventure Land

And now … the mathematics of it:

1. definition of the sine function

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

Figure 3: The sine of an angle

By means of a unit circle (radius r=1) one assigns to each angle \alpha the length of the arc x=x(\alpha) over this angle \alpha. Considering that the circumference of the unit circle is equal to 2\pi, the corresponding radian is x=x(\alpha):

\alphax(\alpha)
0^\circ0
90^\circ\pi/2
180^\circ\pi
270^\circ3\pi/2
360^\circ2\pi
Table 1: Relationship between degrees and radians

The sine function f\colon y=f(x)=\sin(x) is defined by \sin(x)=\sin(x(\alpha)). The length of the so-called adjacency in a right triangle with a hypotenuse of length 1 over the angle \alpha with arc length x=x(\alpha) is called cosine (also: cosine function) \cos(x(\alpha))=\cos(x) with x=x(\alpha).

2. unwinding of the plane section

A (straight) circular cylinder with radius r for the base circle (in the adventure land of mathematics r= 60\mathrm{mm}) is described by the following equations because of the equation \cos^2(\varphi)+\sin^2(\varphi)=1 (Pythagorean theorem at the unit circle):

    \[\begin{pmatrix} x(\varphi)\ y(\varphi)\ z(\varphi)\end{pmatrix}=\begin{pmatrix} r\cos(\varphi)\ r\sin(\varphi)\ z\end{pmatrix}\quad(1).\]

Here r and \varphi are the so-called polar coordinates as shown in the following figure 4:

Figure 4: Polar coordinates

A plane section of the circular cylinder (at the angle of 45^\circ) is given by the bisector in the yz-plane

    \[y=z\quad (2)\]

given. Thus, from equations (1) and (2) it follows

    \[z=r\sin(\varphi)\]

for -\pi\leq\varphi\leq\pi. This sine function is visible with r=60\mathrm{mm} on the foil to be unrolled at the corresponding exhibit in Mathematics Adventure land.


Literature

[1] https://de.wikipedia.org/wiki/Sinus_und_Kosinus

[2] https://de.wikipedia.org/wiki/Zylinder_(Geometrie)