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

Deux stations d’observation, séparées par une distance horizontale connue, observent un satellite à des angles d’élévation connus.La distance entre la première station et le satellite est connue.Le but est de trouver la distance entre le satellite et la deuxième station en résolvant le triangle formé par les deux stations et le satellite.Lorsqu’un côté et deux angles sont connus, ou que deux côtés et un angle opposé à l’un d’eux sont fournis, la loi des sinus est applicable.Selon la loi des sinus, le rapport entre la longueur d’un côté et le sinus de son angle opposé reste constant tout au long d’un triangle.Dans ce cas, un côté c et deux angles C et B sont connus. Cela correspond à la loi des sinus, permettant le calcul du côté b.Prenons un autre exemple. La loi des sinus peut être utilisée pour déterminer la distance entre deux drapeaux sur les rives opposées d’une rivière.Ici, le côté BC est de 100 mètres, l’angle opposé A de 30 degrés. Un autre angle connu est B de 70 degrés, opposé au côté inconnu AC.En utilisant la loi des sinus, BC avec l’angle A est utilisé pour trouver AC avec l’angle B.Ce résultat est la distance oblique entre les deux drapeaux le long de la climatisation latérale.

la loi des sinus

The Law of Sines

La r&#233;solution des triangles obliques, d&#233;pourvus d&#8217;angle droit, repose sur des relations trigonom&#233;triques sp&#233;cifiques, notamment la loi des sinus. En effet, &#224; la diff&#233;rence des triangles rectangles, pour lesquels le th&#233;or&#232;me de Pythagore permet de relier les longueurs des c&#244;t&#233;s, les triangles obliques n&#8217;ont pas d&#8217;angle de 90&#176; et requi&#232;rent donc une approche diff&#233;rente. Le th&#233;or&#232;me de Pythagore n&#8217;est valable que pour les triangles rectangles, ce qui le rend inadapt&#233; &#224; la r&#233;solution des triangles non rectangles. Dans ces cas, la loi des sinus devient essentielle, car elle fournit une relation trigonom&#233;trique valable quelle que soit la mesure des angles.La loi des sinus &#233;tablit un lien direct entre les c&#244;t&#233;s et les angles d&#8217;un triangle, permettant de d&#233;terminer des longueurs ou des angles inconnus d&#232;s lors que l&#8217;on dispose d&#8217;&#233;l&#233;ments suffisants. Son utilit&#233; s&#8217;&#233;tend &#224; diverses disciplines scientifiques et techniques, notamment la navigation, la topographie et l&#8217;astronomie, o&#249; la mesure directe de tous les &#233;l&#233;ments d&#8217;un triangle peut ne pas &#234;tre possible.La loi des sinus &#233;nonce que, dans tout triangle ABC, les rapports des longueurs des c&#244;t&#233;s a, b et c, respectivement oppos&#233;s aux angles A, B et C, aux sinus des angles qui leur sont oppos&#233;s sont &#233;gaux. Cette relation s&#8217;exprime math&#233;matiquement ainsi :<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>Cette relation est valable quelle que soit l&#8217;orientation du triangle ou les longueurs de ses c&#244;t&#233;s. Elle est particuli&#232;rement efficace lorsque deux angles et un c&#244;t&#233; quelconque sont connus, ou lorsque deux c&#244;t&#233;s et l&#8217;angle oppos&#233; &#224; l&#8217;un d&#8217;eux sont fournis.La loi des sinus simplifie la d&#233;termination des mesures inconnues dans les triangles obliques, rationalisant ainsi les calculs dans les applications pratiques.

La loi des sinus

La r&#233;solution des triangles obliques, d&#233;pourvus d&#8217;angle droit, repose sur des relations trigonom&#233;triques sp&#233;cifiques, notamment ...

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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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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 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.

math fundamentals

Solving Oblique Triangles with Sines

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