Name: unit circle trig identities and sum formulas
Uploaded: 2025-11-26T05:44:29.000Z
Duration: 1 min 27 s
Channel: JoVE

Eine trigonometrische Identität ist eine Gleichung mit trigonometrischen Funktionen, die für alle Winkel gilt.Eine solche Identität, die sogenannte pythagoreische Identität, kann mit Hilfe eines Einheitskreises abgeleitet werden.Jeder Punkt auf diesem Kreis entspricht einem Winkel, der von der positiven x-Achse aus gemessen wird.Die x-Koordinate des Punktes ist der Kosinus des Winkels, und die y-Koordinate ist der Sinus des Winkels.Die x - und y-Koordinaten bilden die senkrechten Seiten eines rechtwinkligen Dreiecks, wobei der Radius als Hypotenuse gilt.Nach dem Satz des Pythagoras ist das Quadrat der y-Koordinate plus das Quadrat der x-Koordinate gleich dem Quadrat des Radius.Da der Radius eins ist, beweist dies, dass Sinus zum Quadrat plus Kosinus zum Quadrat gleich eins ist.Dies ist als pythagoreische Identität bekannt und gilt für alle realen Blickwinkel.Diese Identität, die sich in zwei Dimensionen ableitet, hat Anwendungen, die sich auf dreidimensionale Systeme erstrecken. In der Astronomie beschreiben Himmelskoordinaten die Position von Sternen auf der Himmelssphäre.Die Identität stellt sicher, dass diese Positionen auf der Kugel konsistent bleiben, was es den Astronomen ermöglicht, Sterne zu verfolgen, während sie sich über den Himmel zu bewegen scheinen.

trigonometrische identitäten i

Trigonometrische Identit&#228;ten sind Gleichungen, die trigonometrische Funktionen in Beziehung setzen und f&#252;r alle Winkel innerhalb ihrer Definitionsbereiche gelten. Eine grundlegende Identit&#228;t ist die pythagoreische Identit&#228;t, die sich direkt aus der Geometrie des Einheitskreises ergibt. F&#252;r jeden Winkel &#952; hat ein Punkt auf dem Einheitskreis die Koordinaten (cos &#952;, sin &#952;). Da der Radius des Kreises gleich 1 ist, folgt aus dem Satz des Pythagoras:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40078/40078_Equation_1_2__editor_40078.svg' alt='40078_Equation_1_2_.svg' /></figure>Diese Identit&#228;t dient als Grundlage f&#252;r die Herleitung weiterer Identit&#228;ten. Durch Division der pythagoreischen Identit&#228;t durch cos ^2&#952; (unter der Annahme, dass cos &#952; &#8800; 0) ergibt sich:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1920/450;' src='https://cloudfront.jove.com/files/ftp_upload/40078/40078_Equation_2_2__editor_40078.svg' alt='40078_Equation_2_2_.svg' /></figure>Die pythagoreische Identit&#228;t spielt auch bei komplexeren Identit&#228;ten eine Rolle, etwa bei der Kosinus-Additionsformel. Eine Herleitung dieser Formel beruht auf geometrischen &#220;berlegungen mit Punkten auf dem Einheitskreis und der Abstandsformel. Dabei enthalten die Koordinaten der Schl&#252;sselpunkte Kombinationen von Sinus- und Kosinuswerten. Bei Anwendung der Abstandsformel und anschlie&#223;ender Vereinfachung reduziert die pythagoreische Identit&#228;t Ausdr&#252;cke mit Quadraten trigonometrischer Terme. Dies f&#252;hrt letztlich zu:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40078/40078_Equation_3_2__editor_40078.svg' alt='40078_Equation_3_2_.svg' /></figure>Ersetzt man t durch &#8722;t in der Additionsformel und nutzt die Tatsache, dass der Kosinus gerade und der Sinus ungerade ist, erh&#228;lt man die Subtraktionsformel:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40078/40078_Equation_4_2__editor_40078.svg' alt='40078_Equation_4_2_.svg' /></figure>Die Formeln f&#252;r die Addition und Subtraktion des Sinus lassen sich zudem mithilfe von Kofunktionsidentit&#228;ten und Winkeltransformationen herleiten:<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/40078/40078_Equation_5_2__editor_40078.svg' alt='40078_Equation_5_2_.svg' /></figure>Schlie&#223;lich ergeben sich die Tangens-Additions- und -Subtraktionsformeln aus den Summenformeln f&#252;r Sinus und Kosinus durch Bildung ihres Quotienten:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1920/450;' src='https://cloudfront.jove.com/files/ftp_upload/40078/40078_Equation_6_editor_40078.svg' alt='40078_Equation_6.svg' /></figure>Diese Identit&#228;ten sind zentrale Werkzeuge f&#252;r die algebraische Vereinfachung, das L&#246;sen trigonometrischer Gleichungen sowie Anwendungen in der Differential- und Integralrechnung.

Trigonometrische Identitäten I

Trigonometrische Identit&#228;ten sind Gleichungen, die trigonometrische Funktionen in Beziehung setzen und f&#252;r alle Winkel innerhalb ihrer ...

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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 Pythagorean identity. This result helps build later trig formulas and shows how one identity can generate others.

The Pythagorean identity also supports the cosine addition formula. One way to derive it uses points on the unit circle and the distance formula. The coordinates of the points involve sine and cosine terms. After simplifying the distance expression, the Pythagorean identity removes square terms and leads to the cosine sum formula.

The subtraction formula for cosine follows from the addition formula. Replacing t with -t and using the even and odd properties of cosine and sine gives the cosine difference identity. The sine addition and subtraction formulas can also be derived through cofunction identities and angle transformations.

The tangent addition and subtraction identities come from the sine and cosine sum formulas. Taking the ratio of the sine and cosine results produces the tangent formulas. These identities are useful for simplifying algebra, solving trigonometric equations, and working in calculus.

Unit Circle Trig Identities and Sum Formulas

unit-circle-trig-identities-and-sum-formulas

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trigonometric identities i

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 Formulas

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key terms and concepts

Learn how the unit circle leads to Pythagorean, sine, cosine, and tangent identities, including addition and subtraction formulas for high school math.

Unit Circle Trig Identities and Sum Formulas | 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

Unit Circle Trig Identities and Sum Formulas

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