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

Stellen Sie sich ein Dreieck mit den Seiten a, b und c vor, wobei jeder Winkel nach dem Scheitelpunkt gegenüber der entsprechenden Seite benannt ist.Der Kosinussatz besagt, dass das Quadrat einer Seite gleich der Summe der Quadrate der beiden anderen Seiten ist, minus dem Doppelten des Produkts dieser Seiten multipliziert mit dem Kosinus des eingeschlossenen Winkels.Sie gilt, wenn zwei Seiten und der enthaltene Winkel bekannt sind. Er gilt auch, wenn alle drei Seiten gegeben sind und der eingeschlossene Winkel mit dem Kehrwert des Kosinus gefunden wird.Wenn der eingeschlossene Winkel ein rechter Winkel ist, wird das Dreieck zu einem rechtwinkligen Dreieck. Der Kosinus von neunzig Grad ist Null. Dadurch wird der Kosinusterm aus der Formel entfernt und sie zum Satz des Pythagoras vereinfacht.In der Landvermessung kommt der Kosinussatz zum Einsatz, wenn eine Seite eines markierten Dreiecks unzugänglich ist, z. B. wenn es quer über ein Gewässer liegt. Die beiden anderen Seiten und der mitgelieferte Winkel werden mit einem Vermessungsgerät vermessen. Ihre Werte werden dann in den Kosinussatz eingefügt, um die Länge der unbekannten Seite ohne direkte Messung zu berechnen.

der kosinussatz

Der Kosinussatz ist ein grundlegendes Ergebnis der Trigonometrie, das die L&#228;ngen der Seiten eines Dreiecks mit dem Kosinus eines seiner Winkel verkn&#252;pft. Er verallgemeinert den Satz des Pythagoras und erm&#246;glicht Berechnungen in nicht rechtwinkligen Dreiecken, in denen die einfachen Beziehungen der rechtwinkligen Geometrie nicht gelten. Die Formel ist besonders n&#252;tzlich, wenn eine Seite oder ein Winkel nicht direkt messbar ist, etwa in der Vermessung, der Navigation und im Ingenieurwesen.F&#252;r ein Dreieck mit den Seiten a, b, c und den gegen&#252;berliegenden Winkeln A, B, C lautet der Kosinussatz:<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>Entsprechende Ausdr&#252;cke f&#252;r die anderen Seiten lauten:<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>Anwendbar ist der Satz insbesondere in zwei F&#228;llen: wenn zwei Seiten und der eingeschlossene Winkel bekannt sind oder wenn alle drei Seiten bekannt sind und ein Winkel bestimmt werden soll.Neben der Landvermessung findet der Kosinussatz Anwendung in der Astronomie zur Entfernungsbestimmung zwischen Himmelsk&#246;rpern aus irdischer Perspektive. Er wird au&#223;erdem in der Robotik zur Bestimmung von Gelenkwinkeln in Armmechanismen sowie in der Physik zur Aufl&#246;sung von Kr&#228;ften verwendet, die unter einem Winkel wirken.Betrachten Sie beispielsweise ein Dreieck mit a = 7 Einheiten, b = 10 Einheiten und dem eingeschlossenen Winkel c = 60&#176;. Unter Anwendung des Kosinussatzes:<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>Dies zeigt, wie der Kosinussatz pr&#228;zise Entfernungsberechnungen auch in komplexen Geometrien erm&#246;glicht.

Der Kosinussatz

Der Kosinussatz ist ein grundlegendes Ergebnis der Trigonometrie, das die L&#228;ngen der Seiten eines Dreiecks mit dem Kosinus eines seiner Winkel ...

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Math Fundamentals

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KEY TERMS AND CONCEPTS

WICHTIGE BEGRIFFE UND KONZEPTE

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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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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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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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...

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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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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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