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

Zwei Beobachtungsstationen, getrennt durch einen bekannten horizontalen Abstand, beobachten einen Satelliten in bekannten Höhenwinkeln.Die Entfernung von der ersten Station zum Satelliten ist bekannt.Ziel ist es, die Entfernung vom Satelliten zur zweiten Station zu ermitteln, indem das Dreieck gelöst wird, das aus den beiden Stationen und dem Satelliten gebildet wird.Wenn eine Seite und zwei Winkel bekannt sind oder zwei Seiten und ein Winkel entgegengesetzt zu einem von ihnen angegeben sind, ist der Sinussatz anwendbar.Nach dem Sinusgesetz bleibt das Verhältnis einer Seitenlänge zum Sinus ihres entgegengesetzten Winkels in einem Dreieck konstant.In diesem Fall sind eine Seite c und zwei Winkel C und B bekannt. Dies entspricht dem Sinusgesetz und ermöglicht die Berechnung von Seite b.Nehmen Sie ein anderes Beispiel. Das Gesetz der Sinus kann verwendet werden, um den Abstand zwischen zwei Flaggen an gegenüberliegenden Ufern eines Flusses zu ermitteln.Hier ist die Seite BC 100 Meter, der gegenüberliegende Winkel A von 30 Grad. Ein weiterer bekannter Winkel ist B von 70 Grad, gegenüber der unbekannten Seite AC.Unter Verwendung des Sinussatzes wird BC mit dem Winkel A verwendet, um AC mit dem Winkel B zu finden.Dieses Ergebnis ist der schräge Abstand zwischen den beiden Fahnen entlang der Seite AC.

der sinussatz

The Law of Sines

Das L&#246;sen schiefwinkliger Dreiecke &#8211; also solcher ohne rechten Winkel &#8211; beruht auf spezifischen trigonometrischen Beziehungen, insbesondere dem Sinussatz. Im Unterschied zu rechtwinkligen Dreiecken, bei denen der Satz des Pythagoras die Seitenl&#228;ngen verkn&#252;pft, fehlt hier der 90&#176;-Winkel, sodass ein anderer Ansatz erforderlich ist. Der Satz des Pythagoras gilt ausschlie&#223;lich f&#252;r rechtwinklige Dreiecke und ist daher f&#252;r nicht rechtwinklige Dreiecke ungeeignet. In solchen F&#228;llen ist der Sinussatz unverzichtbar, da er eine trigonometrische Beziehung liefert, die unabh&#228;ngig von der Gr&#246;&#223;e der Winkel anwendbar ist.Der Sinussatz stellt eine direkte Beziehung zwischen den Seiten und Winkeln eines Dreiecks her und erm&#246;glicht die Bestimmung unbekannter L&#228;ngen oder Winkel, sofern ausreichende Informationen vorliegen. Sein Nutzen erstreckt sich auf zahlreiche wissenschaftliche und technische Disziplinen &#8211; etwa in der Navigation, der Vermessung und der Astronomie &#8211;, in denen eine direkte Messung aller Dreieckselemente oft nicht praktikabel ist.Der Sinussatz besagt, dass in jedem Dreieck ABC das Verh&#228;ltnis der Seitenl&#228;ngen a, b und c, die den Winkeln A, B und C gegen&#252;berliegen, zum Sinus ihrer jeweiligen Gegenwinkel konstant ist. Mathematisch l&#228;sst sich diese Beziehung wie folgt ausdr&#252;cken:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:56.8%;'><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>Diese Beziehung gilt unabh&#228;ngig von Ausrichtung und Skalierung des Dreiecks. Sie ist besonders effektiv, wenn zwei Winkel und eine beliebige Seitenl&#228;nge bekannt sind oder wenn zwei Seiten und ein einer der beiden Seiten gegen&#252;berliegender Winkel gegeben sind.Der Sinussatz erleichtert die Bestimmung unbekannter Gr&#246;&#223;en in nicht rechtwinkligen Dreiecken und unterst&#252;tzt effiziente Berechnungen in praktischen Anwendungen.

Der Sinussatz

Das L&#246;sen schiefwinkliger Dreiecke &#8211; also solcher ohne rechten Winkel &#8211; beruht auf spezifischen trigonometrischen Beziehungen, ...

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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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