@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .

psr:-JL9Z4260-Z
  skos:prefLabel "trigonométrie"@fr, "trigonometry"@en ;
  a skos:Concept ;
  skos:narrower psr:-X5T7H86P-S .

psr:-X5T7H86P-S
  skos:broader psr:-N1RMZJ03-D, psr:-LN44RXKJ-J, psr:-JL9Z4260-Z ;
  skos:altLabel "cosine rule"@en, "cosine formula"@en ;
  skos:prefLabel "law of cosines"@en, "loi des cosinus"@fr ;
  skos:definition """En mathématiques, la loi des cosinus est un théorème de géométrie couramment utilisé en trigonométrie, qui relie dans un triangle la longueur d'un côté à celles des deux autres et au cosinus de l'angle formé par ces deux côtés. Cette loi s'exprime de façon analogue en géométrie plane, sphérique ou hyperbolique. Cette loi généralise le théorème de Pythagore. Les Éléments d'Euclide contenaient déjà une approche géométrique de la généralisation du théorème de Pythagore dans deux cas particuliers : ceux d'un triangle obtusangle et d'un triangle acutangle. Le développement, au Moyen Âge, de la trigonométrie arabo-musulmane permit au théorème d'évoluer dans sa forme et dans sa portée : l'astronome et mathématicien al-Battani généralisa le résultat d'Euclide à la géométrie sphérique au début du Xe siècle, et l'introduction des fonctions trigonométriques permit à Ghiyath al-Kashi, mathématicien de l'école de Samarcande, de mettre le théorème sous une forme utilisable pour la triangulation au cours du XVe siècle. La propriété a été popularisée en occident par François Viète qui l'a vraisemblablement redécouverte indépendamment. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Loi_des_cosinus">https://fr.wikipedia.org/wiki/Loi_des_cosinus</a>)"""@fr, """In trigonometry, the <b>law of cosines</b> (also known as the <b>cosine formula</b> or <b>cosine rule</b>) relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle a,}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>a</mi>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle a,}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f059f053fcf9f421b7c74362cf3bd5ed024e19d1" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.877ex; height:2.009ex;" alt="a,"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle b,}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>b</mi>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle b,}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdb96677ba71b937617ca8751955f884f6306b64" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.644ex; height:2.509ex;" alt="b,"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle c,}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>c</mi>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle c,}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae5e8f9eb465084d3a00a24026b80652b74ef58e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.654ex; height:2.009ex;" alt="c,"></span> opposite respective angles <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\alpha ,}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>α<!-- α --></mi>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\alpha ,}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b2cc8f6d373595f06dcd33f127dadf2b9d5727f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.134ex; height:2.009ex;" alt="\\\\alpha,"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\eta ,}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>β<!-- β --></mi>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\eta ,}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59ab677d974cccb0132cac08bd67fc8ac765627e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.979ex; height:2.509ex;" alt="\\eta ,"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\gamma }">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>γ<!-- γ --></mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\gamma }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="\\\\gamma "></span> (see Fig. 1), the law of cosines states:
<br/>
<br/><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle {\\egin{aligned}c^{2}&amp;=a^{2}+b^{2}-2ab\\\\cos \\\\gamma ,\\\\\\\\a^{2}&amp;=b^{2}+c^{2}-2bc\\\\cos \\\\alpha ,\\\\\\^{2}&amp;=a^{2}+c^{2}-2ac\\\\cos \\eta .\\\\end{aligned}}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.467em 0.467em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<br/>            <mtr>
<br/>              <mtd>
<br/>                <msup>
<br/>                  <mi>c</mi>
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mn>2</mn>
<br/>                  </mrow>
<br/>                </msup>
<br/>              </mtd>
<br/>              <mtd>
<br/>                <mi></mi>
<br/>                <mo>=</mo>
<br/>                <msup>
<br/>                  <mi>a</mi>
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mn>2</mn>
<br/>                  </mrow>
<br/>                </msup>
<br/>                <mo>+</mo>
<br/>                <msup>
<br/>                  <mi>b</mi>
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mn>2</mn>
<br/>                  </mrow>
<br/>                </msup>
<br/>                <mo>−<!-- − --></mo>
<br/>                <mn>2</mn>
<br/>                <mi>a</mi>
<br/>                <mi>b</mi>
<br/>                <mi>cos</mi>
<br/>                <mo>⁡<!-- ⁡ --></mo>
<br/>                <mi>γ<!-- γ --></mi>
<br/>                <mo>,</mo>
<br/>              </mtd>
<br/>            </mtr>
<br/>            <mtr>
<br/>              <mtd>
<br/>                <msup>
<br/>                  <mi>a</mi>
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mn>2</mn>
<br/>                  </mrow>
<br/>                </msup>
<br/>              </mtd>
<br/>              <mtd>
<br/>                <mi></mi>
<br/>                <mo>=</mo>
<br/>                <msup>
<br/>                  <mi>b</mi>
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mn>2</mn>
<br/>                  </mrow>
<br/>                </msup>
<br/>                <mo>+</mo>
<br/>                <msup>
<br/>                  <mi>c</mi>
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mn>2</mn>
<br/>                  </mrow>
<br/>                </msup>
<br/>                <mo>−<!-- − --></mo>
<br/>                <mn>2</mn>
<br/>                <mi>b</mi>
<br/>                <mi>c</mi>
<br/>                <mi>cos</mi>
<br/>                <mo>⁡<!-- ⁡ --></mo>
<br/>                <mi>α<!-- α --></mi>
<br/>                <mo>,</mo>
<br/>              </mtd>
<br/>            </mtr>
<br/>            <mtr>
<br/>              <mtd>
<br/>                <msup>
<br/>                  <mi>b</mi>
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mn>2</mn>
<br/>                  </mrow>
<br/>                </msup>
<br/>              </mtd>
<br/>              <mtd>
<br/>                <mi></mi>
<br/>                <mo>=</mo>
<br/>                <msup>
<br/>                  <mi>a</mi>
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mn>2</mn>
<br/>                  </mrow>
<br/>                </msup>
<br/>                <mo>+</mo>
<br/>                <msup>
<br/>                  <mi>c</mi>
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mn>2</mn>
<br/>                  </mrow>
<br/>                </msup>
<br/>                <mo>−<!-- − --></mo>
<br/>                <mn>2</mn>
<br/>                <mi>a</mi>
<br/>                <mi>c</mi>
<br/>                <mi>cos</mi>
<br/>                <mo>⁡<!-- ⁡ --></mo>
<br/>                <mi>β<!-- β --></mi>
<br/>                <mo>.</mo>
<br/>              </mtd>
<br/>            </mtr>
<br/>          </mtable>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\egin{aligned}c^{2}&amp;=a^{2}+b^{2}-2ab\\\\cos \\\\gamma ,\\\\\\\\a^{2}&amp;=b^{2}+c^{2}-2bc\\\\cos \\\\alpha ,\\\\\\^{2}&amp;=a^{2}+c^{2}-2ac\\\\cos \\eta .\\\\end{aligned}}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd9a9823b7eec03c9da379cfba653a92126cb4ff" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -4.505ex; width:25.423ex; height:10.176ex;" alt="{\\\\displaystyle {\\egin{aligned}c^{2}&amp;=a^{2}+b^{2}-2ab\\\\cos \\\\gamma ,\\\\\\\\a^{2}&amp;=b^{2}+c^{2}-2bc\\\\cos \\\\alpha ,\\\\\\^{2}&amp;=a^{2}+c^{2}-2ac\\\\cos \\eta .\\\\end{aligned}}}"></span></dd></dl>
<br/>The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\gamma }">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>γ<!-- γ --></mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\gamma }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="\\\\gamma "></span> is a right angle then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\cos \\\\gamma =0,}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>cos</mi>
<br/>        <mo>⁡<!-- ⁡ --></mo>
<br/>        <mi>γ<!-- γ --></mi>
<br/>        <mo>=</mo>
<br/>        <mn>0</mn>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\cos \\\\gamma =0,}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0929ad4c1a953508b7831c5920580215a14c85e7" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:9.668ex; height:2.676ex;" alt="{\\\\displaystyle \\\\cos \\\\gamma =0,}"></span> and the law of cosines reduces to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle c^{2}=a^{2}+b^{2}.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msup>
<br/>          <mi>c</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>=</mo>
<br/>        <msup>
<br/>          <mi>a</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>+</mo>
<br/>        <msup>
<br/>          <mi>b</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle c^{2}=a^{2}+b^{2}.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e10f9b628ee334c9ec8dcf1f54db092787bebdc5" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.505ex; width:12.983ex; height:2.843ex;" alt="{\\\\displaystyle c^{2}=a^{2}+b^{2}.}"></span>
<br/>The law of cosines is useful for solving a triangle when all three sides or two sides and their included angle are given. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Law_of_cosines">https://en.wikipedia.org/wiki/Law_of_cosines</a>)"""@en ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Loi_des_cosinus>, <https://en.wikipedia.org/wiki/Law_of_cosines> ;
  skos:related psr:-RX61SX55-G ;
  a skos:Concept ;
  skos:inScheme psr: .

psr: a skos:ConceptScheme .
psr:-N1RMZJ03-D
  skos:prefLabel "hyperbolic triangle"@en, "triangle hyperbolique"@fr ;
  a skos:Concept ;
  skos:narrower psr:-X5T7H86P-S .

psr:-RX61SX55-G
  skos:prefLabel "triangle"@fr, "triangle"@en ;
  a skos:Concept ;
  skos:related psr:-X5T7H86P-S .

psr:-LN44RXKJ-J
  skos:prefLabel "trigonométrie sphérique"@fr, "spherical trigonometry"@en ;
  a skos:Concept ;
  skos:narrower psr:-X5T7H86P-S .