Name: solving oblique triangles with sines
Uploaded: 2025-11-26T05:44:32.000Z
Duration: 1 min 29 s
Channel: JoVE

Duas estações de observação, separadas por uma distância horizontal conhecida, observam um satélite em ângulos de elevação conhecidos.A distância da primeira estação ao satélite é conhecida.O objetivo é encontrar a distância do satélite à segunda estação, resolvendo o triângulo formado pelas duas estações e o satélite.Quando um lado e dois ângulos são conhecidos, ou dois lados e um ângulo oposto a um deles são fornecidos, a Lei dos Senos é aplicável.De acordo com a Lei dos Senos, a razão entre o comprimento de um lado e o seno de seu ângulo oposto permanece constante ao longo de um triângulo.Nesse caso, um lado c e dois ângulos C e B são conhecidos. Isso se encaixa na Lei dos Senos, permitindo o cálculo do lado b.Veja outro exemplo. A Lei dos Senos pode ser usada para encontrar a distância entre duas bandeiras em margens opostas de um rio.Aqui, o lado BC é de 100 metros, ângulo oposto A de 30 graus. Outro ângulo conhecido é B de 70 graus, oposto ao lado desconhecido AC.Usando a Lei dos Senos, BC com o ângulo A é usado para encontrar AC com o ângulo B.Este resultado é a distância inclinada entre as duas bandeiras ao longo do lado AC.

a lei dos senos

The Law of Sines

A resolu&#231;&#227;o de tri&#226;ngulos obl&#237;quos (aqueles sem &#226;ngulos retos) depende de rela&#231;&#245;es trigonom&#233;tricas espec&#237;ficas, notadamente a Lei dos Senos. Isso ocorre porque, diferentemente dos tri&#226;ngulos ret&#226;ngulos, em que o teorema de Pit&#225;goras pode ser usado para relacionar os comprimentos dos lados, os tri&#226;ngulos obl&#237;quos n&#227;o possuem um &#226;ngulo de 90&#176;, e portanto requerem uma abordagem diferente. O teorema de Pit&#225;goras &#233; v&#225;lido apenas para tri&#226;ngulos ret&#226;ngulos, tornando-o inadequado para a resolu&#231;&#227;o de tri&#226;ngulos obl&#237;quos. Nesses casos, a Lei dos Senos se torna essencial, pois fornece uma rela&#231;&#227;o trigonom&#233;trica que funciona independentemente da medida do &#226;ngulo.A Lei dos Senos fornece uma conex&#227;o direta entre os lados e os &#226;ngulos de um tri&#226;ngulo, permitindo a determina&#231;&#227;o de comprimentos ou &#226;ngulos desconhecidos quando h&#225; informa&#231;&#245;es suficientes. Sua utilidade se estende a diversas disciplinas cient&#237;ficas e t&#233;cnicas, abrangendo navega&#231;&#227;o, topografia e astronomia, onde a medi&#231;&#227;o direta de todos os elementos de um tri&#226;ngulo pode n&#227;o ser vi&#225;vel.A Lei dos Senos afirma que, em qualquer tri&#226;ngulo ABC, as raz&#245;es entre os comprimentos dos lados a, b e c, opostos aos &#226;ngulos A, B e C, respectivamente, pelo seno dos &#226;ngulos opostos correspondentes, s&#227;o constantes. Essa rela&#231;&#227;o &#233; expressa matematicamente como:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/40077/40077_Equation_1_1__editor_40077.svg' alt='40077_Equation_1_1_.svg' /></figure>Essa rela&#231;&#227;o &#233; v&#225;lida independentemente da orienta&#231;&#227;o ou dos comprimentos dos lados do tri&#226;ngulo. Ela &#233; especialmente eficaz nos casos em que dois &#226;ngulos e qualquer lado s&#227;o conhecidos, ou quando dois lados e um &#226;ngulo oposto a um deles s&#227;o dados.A Lei dos Senos simplifica o processo de resolu&#231;&#227;o de medidas desconhecidas em tri&#226;ngulos n&#227;o ret&#226;ngulos, agilizando os c&#225;lculos em aplica&#231;&#245;es pr&#225;ticas.

A Lei dos Senos

A resolu&#231;&#227;o de tri&#226;ngulos obl&#237;quos (aqueles sem &#226;ngulos retos) depende de rela&#231;&#245;es trigonom&#233;tricas ...

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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 unknown sides or unknown angles when enough information is known. It is especially useful when two angles and any one side are given. It also works when two sides and the angle opposite one of them are known.

The Law of Sines is useful in navigation, surveying, and astronomy. In these fields, direct measurement of every part of a triangle may not be possible. The law makes it easier to solve non-right triangles and find missing measurements.

Solving Oblique Triangles with Sines

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the law of 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...

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key terms and concepts

Learn how the Law of Sines helps solve oblique triangles by linking sides and angles. Useful for finding missing measurements in non-right triangles.

Solving Oblique Triangles with Sines | 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.

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Solving Oblique Triangles with Sines

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