Sine

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) $\sin(\alpha)$ of an angle $\alpha$ is the length of the so-called opposite cathetus in a right triangle with a hypotenuse of length one.

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 $x=x(\alpha)$ :

$\alpha$	$x(\alpha)$
$0^\circ$	$0$
$90^\circ$	$\pi/2$
$180^\circ$	$\pi$
$270^\circ$	$3\pi/2$
$360^\circ$	$2\pi$

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 adjacent cathetus in a right triangle with a hypotenuse of length 1 over the angle $\alpha$ with the arc length $x=x(\alpha)$ is called the cosine (also: cosine function) $\cos(x(\alpha))=\cos(x)$ with $x=x(\alpha)$ .

2. unwinding of the plane cut

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

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

$\begin{align*}y=z\quad (2) .\end{align*}$

From equations (1) and (2) thus follows

$z=r\sin(\varphi)$

for $-\pi\leq\varphi\leq\pi$ . This sine function is visible with $r=60\mathrm {mm}$ on the slide to be unrolled at the corresponding exhibit in the ADVENTURELAND MATHEMATICS.

Literature

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

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