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

Considere um triângulo com lados a, b e c, onde cada ângulo é nomeado após o vértice oposto ao lado correspondente.A Lei dos Cosenos afirma que o quadrado de um lado é igual à soma dos quadrados dos outros dois lados, menos o dobro do produto desses lados multiplicado pelo cosseno do ângulo incluído.Aplica-se quando os dois lados e o ângulo incluído são conhecidos. Também se aplica quando todos os três lados são dados e o ângulo incluído é encontrado usando o inverso do cosseno.Quando o ângulo incluído é um ângulo reto, o triângulo se torna um triângulo retângulo. O cosseno de noventa graus é zero. Isso remove o termo cosseno da fórmula e o simplifica para o Teorema de Pitágoras.Na topografia, a Lei dos Cossenos é usada quando um lado de um triângulo marcado é inacessível, como quando se encontra em um corpo d'água. Os outros dois lados e o ângulo incluído são medidos usando equipamentos de topografia. Seus valores são então substituídos pela Lei dos Cossenos para calcular o comprimento do lado desconhecido sem medição direta.

a lei dos cossenos

A Lei dos Cossenos &#233; um resultado fundamental na trigonometria que relaciona os comprimentos dos lados de um tri&#226;ngulo ao cosseno de um de seus &#226;ngulos. Ela funciona como uma generaliza&#231;&#227;o do Teorema de Pit&#225;goras, possibilitando c&#225;lculos em tri&#226;ngulos n&#227;o ret&#226;ngulos, em que as rela&#231;&#245;es simples da geometria retangular n&#227;o se aplicam. A f&#243;rmula &#233; especialmente &#250;til em situa&#231;&#245;es nas quais a medi&#231;&#227;o direta de um lado ou de um &#226;ngulo n&#227;o &#233; vi&#225;vel, como em medi&#231;&#245;es topogr&#225;ficas, navega&#231;&#227;o e engenharia.Para qualquer tri&#226;ngulo com lados a, b e c, e &#226;ngulos opostos correspondentes A, B e C, a Lei dos Cossenos &#233; dada por:<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>Express&#245;es semelhantes podem ser escritas para os outros 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>Essa lei &#233; aplic&#225;vel em duas situa&#231;&#245;es principais: quando dois lados e o &#226;ngulo compreendido s&#227;o conhecidos, ou quando todos os tr&#234;s lados s&#227;o conhecidos e o &#226;ngulo compreendido precisa ser determinado.Al&#233;m da topografia, a Lei dos Cossenos &#233; utilizada na astronomia para calcular dist&#226;ncias entre corpos celestes que formam tri&#226;ngulos a partir da perspectiva da Terra. Tamb&#233;m &#233; aplicada na rob&#243;tica para determinar &#226;ngulos de articula&#231;&#227;o em mecanismos de bra&#231;o e na f&#237;sica para decompor for&#231;as que atuam em &#226;ngulos.Por exemplo, considere um tri&#226;ngulo em que os lados a = 7 unidades, b = 10 unidades e o &#226;ngulo compreendido c = 60&#186;. Aplicando a Lei dos Cossenos:<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>Isso demonstra como a Lei dos Cossenos permite c&#225;lculos precisos de dist&#226;ncias at&#233; mesmo em geometrias complexas.

A Lei dos Cossenos

A Lei dos Cossenos &#233; um resultado fundamental na trigonometria que relaciona os comprimentos dos lados de um tri&#226;ngulo ao cosseno de um de seus ...

Education

JoVE Core

Math Fundamentals

Fundamentos da Matemática

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Unidade

Trigonometry

Trigonometria

KEY TERMS AND CONCEPTS

TERMOS E CONCEITOS-CHAVE

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

radian and degree measure

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Degrees and radians are the two main units used to measure angles. They both describe rotation around a fixed point. A full turn equals 360 degrees or 2π radians, depending on the unit used. Degrees divide a circle into 360 equal parts. This makes them easy to use in everyday math. They are common in navigation, design, and basic geometry. Radians come from the geometry of a circle. One radian is the angle made when the arc length matches the radius of the circle. Since a circle’s...

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The unit circle is the basis for trigonometric functions on real numbers. It is a circle with radius one centered at the origin of the coordinate plane. On this circle, arc length means the distance measured along the circumference between two points. Each real number t is shown as an arc length measured counterclockwise from the positive x-axis. A point on the unit circle has coordinates (x, y). These coordinates define cosine and sine. The x-coordinate equals cos t, and the y-coordinate...

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Trigonometric Wave Graphs and Asymptotes

Trigonometric functions create repeating waves and curved graphs with clear symmetry. Their shapes come from the unit circle, where a point moves counterclockwise and its vertical and horizontal positions form the sine and cosine values. Sine and cosine graphs are smooth waves with the same period, or repeat length. Their ranges stay bounded between -1 and 1. Sine, cosine, and the other trig functions all show regular patterns, but each one has its own graph shape. Tangent is the ratio of...

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Trig Ratios for Right Triangle Sides

Right triangle trigonometry uses sine, cosine, and tangent to connect an acute angle with the sides of the triangle. These trig ratios describe how side lengths relate to one another in right-angled geometry. They give a clear way to measure the links between angles and sides. Sine compares the side opposite an angle to the hypotenuse. Cosine compares the side adjacent to the angle to the hypotenuse. Tangent compares the opposite side to the adjacent side. Each ratio depends on the angle, not...

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Measuring rocket height with trigonometry uses a fixed ground point and a changing angle of elevation. An observer stands a known horizontal distance from the launch site and watches the object rise. The angle between the ground and the line of sight gives the key information for finding height. The tangent function is the main tool for this calculation. Tangent is the ratio of the vertical side to the adjacent side in a right triangle. When the ground distance is fixed, the object’s height...

the law of sines

Solving Oblique Triangles with Sines

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...

trigonometric identities i

Unit Circle Trig Identities and Sum Formulas

Trigonometric identities link trigonometric functions and stay true for every angle in their domains. A key example is the Pythagorean identity, which comes directly from the geometry of the unit circle. For an angle θ, a point on the unit circle has coordinates (cos θ, sin θ). Because the radius is 1, the Pythagorean Theorem gives the basic identity. That same identity can be rearranged to create another useful form. If it is divided by cos²θ, assuming cos θ is not 0, it leads to a second...

trigonometric identities ii

Double-Angle and Half-Angle Formulas

Double-angle and half-angle trigonometric formulas connect angles that are multiplied or divided by two. They come from the sum and difference formulas, so they build on earlier trigonometry skills. These identities help simplify expressions, solve equations, and evaluate integrals. The double-angle formulas come from substituting equal angles into the sum formulas. For sine and cosine, they link the value at 2θ to the value at θ. The tangent double-angle identity follows the same idea. These...

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Cofunction Identities and Complementary Angles

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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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