{"id":3798,"date":"2022-09-15T14:58:56","date_gmt":"2022-09-15T12:58:56","guid":{"rendered":"https:\/\/erlebnisland-mathematik.de\/?page_id=3798"},"modified":"2023-01-26T14:06:27","modified_gmt":"2023-01-26T13:06:27","slug":"advanced-text-sine","status":"publish","type":"page","link":"https:\/\/erlebnisland-mathematik.de\/en\/advanced-text-sine\/","title":{"rendered":"Advanced text Sine"},"content":{"rendered":"

[vc_row drowwidth=”sidebar-biest-default sidebar-biest”][vc_column][vc_column_text]<\/p>\n

Sine<\/h1>\n

[\/vc_column_text][vc_single_image image=”1761″ img_size=”large” alignment=”center”]Figure 1: Poultry sausage. Where is there a sine here?[\/vc_single_image][vc_column_text] As this picture of a sliced poultry sausage shows — and as we see it every day “at the butcher’s<\/em>” — sausages are very often cut at an angle. Geometry teaches us that the intersection is then not bounded by a circle<\/em> but by an ellipse<\/em>. If one cuts the sausage shell parallel to the “main axis<\/em>” of the sausage and places it on a flat surface, the initially “spatial cut<\/em>” becomes a flat curve that represents part of a so-called sine curve<\/em>, i.e. a harmonic oscillation<\/em>.<\/p>\n

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<\/em> 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<\/em>.[\/vc_column_text][vc_single_image image=”1765″ img_size=”large” alignment=”center”]Figure 2: Exhibit in the Maths Adventure Land[\/vc_single_image][vc_column_text]<\/p>\n

And now … the mathematics:<\/h3>\n

[\/vc_column_text][vc_column_text]<\/p>\n

1. definition of the sine function<\/h4>\n

According to the following figure 3, the sine (also: sine function<\/em>) \"\sin(\alpha)\" of an angle \"\alpha\" is the length of the so-called opposite cathetus<\/em> in a right triangle<\/em> with a hypotenuse<\/em> of length one.[\/vc_column_text][vc_single_image image=”1769″ img_size=”large” alignment=”center”]Figure 3: The sine of an angle[\/vc_single_image][vc_column_text]By means of a unit circle<\/em> (radius<\/em> \"r=1\") one assigns to each angle \"\alpha\" the length of the arc \"x=x(\alpha)\" over this angle \"\alpha\". Considering that the circumference<\/em> of the unit circle is equal to \"2\pi\", the corresponding radian is \"x=x(\alpha)\":[\/vc_column_text][vc_column_text]\n\n\n\n\t\n\n\t\n\t\n\t\n\t\n\t
\"\alpha\"<\/th>\"x(\alpha)\"<\/th>\n<\/tr>\n<\/thead>\n
\"0^\circ\"<\/td>\"0\"<\/td>\n<\/tr>\n
\"90^\circ\"<\/td>\"\pi/2\"<\/td>\n<\/tr>\n
\"180^\circ\"<\/td>\"\pi\"<\/td>\n<\/tr>\n
\"270^\circ\"<\/td>\"3\pi/2\"<\/td>\n<\/tr>\n
\"360^\circ\"<\/td>\"2\pi\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/p>\n

Table 1: Relationship between degrees and radians<\/p>\n

[\/vc_column_text][vc_column_text]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<\/em> in a right triangle with a hypotenuse<\/em> of length 1 over the angle \"\alpha\" with the arc length<\/em> \"x=x(\alpha)\" is called the cosine<\/em> (also: cosine function<\/em>) \"\cos(x(\alpha))=\cos(x)\" with \"x=x(\alpha)\".<\/p>\n

2. unwinding of the plane cut<\/h4>\n

A (straight) circular cylinder<\/em> with radius \"r\" for the base circle<\/em> (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<\/em> on the unit circle): <\/p>\n

  <\/span>   <\/span>\"\[\begin {pmatrix} x(\varphi)\\ y(\varphi)\\ z(\varphi)\end {pmatrix} =\begin {pmatrix} r\cos(\varphi)\\ r\sin(\varphi)\\ z\end {pmatrix} \quad(1).\]\"<\/p>\n

Here \"r\" and \"\varphi\" are the so-called polar coordinates<\/em>, as shown in the following figure 4:[\/vc_column_text][vc_single_image image=”1773″ img_size=”large” alignment=”center”]Figure 4: Polar coordinates[\/vc_single_image][vc_column_text]A plane section of the circular cylinder (at the angle of \"45^\circ\") is given by the bisector<\/em> in the \"yz\"-plane <\/p>\n

  <\/span>   <\/span>\"\begin{align*}y=z\quad (2)<\/p>\n

From equations (1) and (2) thus follows <\/p>\n

  <\/span>   <\/span>\"\[z=r\sin(\varphi)\]\"<\/p>\n

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.[\/vc_column_text][vc_column_text]<\/p>\n

Literature<\/h3>\n

[\/vc_column_text][vc_column_text][1] https:\/\/de.wikipedia.org\/wiki\/Sinus_und_Kosinus<\/a><\/p>\n

[2] https:\/\/de.wikipedia.org\/wiki\/Zylinder_(Geometrie)<\/a>[\/vc_column_text][\/vc_column][\/vc_row]<\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"

[vc_row drowwidth=”sidebar-biest-default sidebar-biest”][vc_column][vc_column_text] Sine [\/vc_column_text][vc_single_image image=”1761″ img_size=”large” alignment=”center”]Figure 1: Poultry sausage. Where is there a sine here?[\/vc_single_image][vc_column_text] 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 … Advanced text Sine<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"footnotes":""},"folder":[],"class_list":["post-3798","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/pages\/3798","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/comments?post=3798"}],"version-history":[{"count":23,"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/pages\/3798\/revisions"}],"predecessor-version":[{"id":4665,"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/pages\/3798\/revisions\/4665"}],"wp:attachment":[{"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/media?parent=3798"}],"wp:term":[{"taxonomy":"folder","embeddable":true,"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/folder?post=3798"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}