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

Considere un triángulo con lados a, b y c, donde cada ángulo lleva el nombre del vértice opuesto a su lado correspondiente.La Ley de los Cosenos establece que el cuadrado de un lado es igual a la suma de los cuadrados de los otros dos lados, menos el doble del producto de esos lados multiplicado por el coseno del ángulo incluido.Se aplica cuando se conocen dos lados y el ángulo incluido. También se aplica cuando se dan los tres lados y el ángulo incluido se encuentra usando el inverso del coseno.Cuando el ángulo incluido es un ángulo recto, el triángulo se convierte en un triángulo rectángulo. El coseno de noventa grados es cero. Esto elimina el término coseno de la fórmula y lo simplifica al teorema de Pitágoras.En la topografía, la Ley de Cosenos se usa cuando un lado de un triángulo marcado es inaccesible, como cuando se encuentra a través de un cuerpo de agua. Los otros dos lados y el ángulo incluido se miden utilizando equipos topográficos. Sus valores se sustituyen en la Ley de Cosenos para calcular la longitud del lado desconocido sin medición directa.

la ley de los cosenos

La ley de los cosenos es un resultado fundamental en trigonometr&#237;a que relaciona las longitudes de los lados de un tri&#225;ngulo con el coseno de uno de sus &#225;ngulos. Constituye una generalizaci&#243;n del teorema de Pit&#225;goras, permitiendo c&#225;lculos en tri&#225;ngulos no rect&#225;ngulos donde las relaciones simples de la geometr&#237;a de los tri&#225;ngulos rect&#225;ngulos ya no son aplicables. La f&#243;rmula es especialmente &#250;til en situaciones en las cuales la medici&#243;n directa de un lado o un &#225;ngulo no es factible, como en aplicaciones de topograf&#237;a, navegaci&#243;n e ingenier&#237;a.Para cualquier tri&#225;ngulo con lados a, b y c, y &#225;ngulos opuestos correspondientes A, B y C, la ley de los cosenos se expresa mediante:<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>Se pueden escribir expresiones similares para los dem&#225;s lados:<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>Esta ley se aplica en dos situaciones principales: cuando se conocen dos lados y el &#225;ngulo comprendido, o cuando se conocen los tres lados y ha de determinarse el &#225;ngulo comprendido.Adem&#225;s de la topograf&#237;a terrestre, la ley de los cosenos se aplica en astronom&#237;a para calcular distancias entre cuerpos celestes que forman tri&#225;ngulos desde la perspectiva terrestre. Tambi&#233;n se utiliza en rob&#243;tica para determinar &#225;ngulos de articulaci&#243;n en mecanismos de brazos y en f&#237;sica para resolver fuerzas que act&#250;an en &#225;ngulo.Por ejemplo, consid&#233;rese un tri&#225;ngulo donde los lados a = 7 unidades, b = 10 unidades y el &#225;ngulo comprendido C = 60&#176;. Usando la ley de los cosenos:<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>Esto demuestra c&#243;mo la ley de los cosenos permite c&#225;lculos precisos de distancias incluso en geometr&#237;as complejas.

La ley de los cosenos

La ley de los cosenos es un resultado fundamental en trigonometr&#237;a que relaciona las longitudes de los lados de un tri&#225;ngulo con el coseno de ...

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Math Fundamentals

Fundamentos de matemáticas

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Unidad

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Trigonometría

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

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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Solving oblique triangles uses the Law of Sines. An oblique triangle is a triangle with no right angle. These triangles need a different trigonometric method because the Pythagorean theorem works only for right triangles. The Law of Sines links each side of a triangle to the sine of its opposite angle. In triangle ABC, side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. The ratios a/sin A, b/sin B, and c/sin C are constant. This relationship helps find...

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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