Name: using cosines to find triangle sides
Uploaded: 2025-11-26T05:44:31.000Z
Duration: 1 min 15 s
Channel: JoVE

Considérons un triangle de côtés a, b et c, où chaque angle est nommé d’après le sommet opposé à son côté correspondant.La loi des cosinus stipule que le carré d’un côté est égal à la somme des carrés des deux autres côtés, moins deux fois le produit de ces côtés multiplié par le cosinus de l’angle inclus.Il s’applique lorsque les deux côtés et l’angle inclus sont connus. Cela s’applique également lorsque les trois côtés sont donnés et que l’angle inclus est déterminé en utilisant l’inverse du cosinus.Lorsque l’angle inclus est un angle droit, le triangle devient un triangle rectangle. Le cosinus de quatre-vingt-dix degrés est nul. Cela supprime le terme cosinus de la formule et le simplifie en théorème de Pythagore.En arpentage, la loi des cosinus est utilisée lorsqu’un côté d’un triangle marqué est inaccessible, par exemple lorsqu’il se trouve en travers d’un plan d’eau. Les deux autres côtés et l’angle inclus sont mesurés à l’aide d’un appareil d’arpentage. Leurs valeurs sont ensuite substituées dans la loi des cosinus pour calculer la longueur du côté inconnu sans mesure directe.

la loi des cosinus

La loi des cosinus est un r&#233;sultat fondamental de la trigonom&#233;trie qui relie les longueurs des c&#244;t&#233;s d&#8217;un triangle au cosinus de l&#8217;un de ses angles. Elle g&#233;n&#233;ralise le th&#233;or&#232;me de Pythagore et permet des calculs dans les triangles non rectangles, o&#249; les relations simples propres aux triangles rectangles ne s&#8217;appliquent plus. Cette formule est particuli&#232;rement utile lorsque la mesure directe d&#8217;un c&#244;t&#233; ou d&#8217;un angle n&#8217;est pas possible, notamment en topographie, en navigation et en ing&#233;nierie.Pour tout triangle de c&#244;t&#233;s a, b et c et d&#8217;angles oppos&#233;s A, B et C, la loi des cosinus s&#8217;&#233;crit&#160;:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/41304/41304_Equation_1_1__editor_41304.svg' alt='41304_Equation_1_1_.svg' /></figure>Des expressions analogues s&#8217;&#233;crivent pour les autres c&#244;t&#233;s :<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/41304/41304_Equation_2_1__editor_41304.svg' alt='41304_Equation_2_1_.svg' /></figure><figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/41304/41304_Equation_3_1__editor_41304.svg' alt='41304_Equation_3_1_.svg' /></figure>Cette loi s&#8217;applique principalement dans deux cas : lorsque deux c&#244;t&#233;s et l&#8217;angle compris sont connus, ou lorsque les trois c&#244;t&#233;s sont connus et que l&#8217;on cherche l&#8217;angle compris.Au-del&#224; de la topographie, la loi des cosinus trouve des applications en astronomie pour calculer des distances entre corps c&#233;lestes vus depuis la Terre. Elle est &#233;galement utilis&#233;e en robotique pour d&#233;terminer les angles d&#8217;articulation d&#8217;un bras robotis&#233;, et en physique pour d&#233;composer des forces agissant selon des directions inclin&#233;es.Par exemple, consid&#233;rons un triangle de c&#244;t&#233;s a = 7 unit&#233;s, b = 10 unit&#233;s et d&#8217;angle compris c = 60&#176;. En appliquant la loi des cosinus&#160;:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/41304/41304_Equation_4_1__editor_41304.svg' alt='41304_Equation_4_1_.svg' /></figure><figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/41304/41304_Equation_5_1__editor_41304.svg' alt='41304_Equation_5_1_.svg' /></figure><figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/41304/41304_Equation_6_1__editor_41304.svg' alt='41304_Equation_6_1_.svg' /></figure>On voit ainsi comment la loi des cosinus permet des calculs pr&#233;cis de distances, m&#234;me dans des g&#233;om&#233;tries complexes.

La loi des cosinus

La loi des cosinus est un r&#233;sultat fondamental de la trigonom&#233;trie qui relie les longueurs des c&#244;t&#233;s d&#8217;un triangle au cosinus de ...

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Math Fundamentals

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KEY TERMS AND CONCEPTS

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The Law of Cosines

using cosines to find triangle sides

The Law of Cosines helps find unknown sides and angles in any triangle. It is a key trig formula for non-right triangles, where the Pythagorean Theorem does not give a direct answer. The law links side lengths to the cosine of an angle, so students can solve triangles with missing measurements.

For a triangle with sides a, b, and c and opposite angles A, B, and C, the formula is written as a squared equals b squared plus c squared minus 2bc cosine A. Similar equations can be used for the other sides. This makes the Law of Cosines useful when two sides and the included angle are known, or when all three sides are known and the missing angle must be found.

The law is often used when direct measurement is hard. Surveyors can use it for land measurements, and it can help in navigation and engineering. It also appears in astronomy to find distances between celestial bodies when they form triangles from Earth’s point of view.

The Law of Cosines also has uses beyond geometry class. Robotics uses it to determine joint angles in arm mechanisms, and physics uses it to resolve forces acting at angles. For example, if a triangle has sides a = 7 units and b = 10 units with an included angle C = 60 degrees, the law can be used to calculate the remaining side. This shows how cosine relationships help solve real triangle problems with precision.

Using Cosines to Find Triangle Sides

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the law of cosines

The Law of Cosines helps find unknown sides and angles in any triangle. It is a key trig formula for non-right triangles, where the Pythagorean Theorem does not give a direct answer. The law links side lengths to the cosine of an angle, so students can solve triangles with missing measurements. For a triangle with sides a, b, and c and opposite angles A, B, and C, the formula is written as a squared equals b squared plus c squared minus 2bc cosine A. Similar equations can be used for the other...

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key terms and concepts

Learn how the Law of Cosines helps solve non-right triangles, find missing sides or angles, and supports surveying, navigation, engineering, and more.

Using Cosines to Find Triangle Sides | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Using Cosines to Find Triangle Sides

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