Name: inverse trig derivatives in tracking
Uploaded: 2026-01-12T12:24:15.000Z
Duration: 1 min 30 s
Channel: JoVE

Ein Schiff verfolgt ein herannahendes Flugzeug, indem es die Schrägentfernung und den Höhenwinkel misst. Aus diesen Messungen lassen sich die horizontalen und vertikalen Komponenten der Entfernung mittels trigonometrischer Beziehungen bestimmenUm die Änderung des Winkels mathematisch zu analysieren, wird der Winkel Theta als y abgebildet und das Höhen-zu-Entfernungs-Verhältnis als x, was die Relation y als Arcsin x ergibt.Das Maß, wie momentane Veränderungen in x y beeinflussen, ist die Winkelempfindlichkeit, und dies erfordert implizite Differenzierung.Der Sinus beider Seiten der Gleichung liefert eine äquivalente trigonometrische Form, ausgedrückt als Sinus y gleich x.Diese Gleichung wird dann implizit bezüglich x differenziert, um die Ableitung zu finden.Durch Umanordnung und Anwendung der pythagoreischen Identität kann die Kosinusfunktion in Sinusform geschrieben werden. Da der Sinus von y gleich x ist, kann dies zurückgesetzt werden, um die Ableitung von Arcsin von x zu erhalten.Wenn das Flugzeug über dem Kopf ist, verursacht eine winzige Änderung von x eine enorme Veränderung des Winkels, was zu einer Explosion der Verfolgungsempfindlichkeit und anschließendem Verfolgungsfehler führt. Ebenso können auch die Ableitungen der verbleibenden inversen trigonometrischen Funktionen gefunden werden.

ableitungen inverser trigonometrischer funktionen

Derivatives of Inverse Trigonometric Functions

Ein Schiff, das ein sich n&#xE4;herndes Flugzeug verfolgt, nutzt geometrische Messungen, um dessen Position relativ zum Beobachter zu bestimmen. Durch die Messung der Schr&#xE4;gentfernung zum Flugzeug und des H&#xF6;henwinkels lassen sich die horizontalen und vertikalen Komponenten der Entfernung mithilfe trigonometrischer Beziehungen ermitteln. Dieser geometrische Ansatz bildet die Grundlage f&#xFC;r die Analyse, wie sich der beobachtete Winkel ver&#xE4;ndert, w&#xE4;hrend sich das Flugzeug dem Schiff n&#xE4;hert.Zur Untersuchung des mathematischen Verhaltens des H&#xF6;henwinkels wird dieser als abh&#xE4;ngige Variable modelliert, w&#xE4;hrend das Verh&#xE4;ltnis der Flugh&#xF6;he zur Schr&#xE4;gentfernung als unabh&#xE4;ngige Variable dient. Diese Zuordnung f&#xFC;hrt zu einer inversen trigonometrischen Beziehung zwischen den beiden Variablen, die die Basis f&#xFC;r die Analyse der Nachf&#xFC;hrungsempfindlichkeit bildet. Die Empfindlichkeit beschreibt, wie stark kleine &#xC4;nderungen des Verh&#xE4;ltnisses den gemessenen Winkel beeinflussen, und wird durch Differentiation quantifiziert.<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:25%;"><img style="aspect-ratio:300/24;" src="https://cloudfront.jove.com/files/ftp_upload/19805/19805_Equation_1_editor_19805.svg" alt="19805_Equation_1.svg" /></figure>Zur Differentiation dieser inversen Beziehung wird sie zun&#xE4;chst durch Anwendung der Sinusfunktion auf beide Seiten in eine &#xE4;quivalente trigonometrische Form &#xFC;berf&#xFC;hrt. Diese Transformation erm&#xF6;glicht die Anwendung der impliziten Differentiation, die erforderlich ist, da die abh&#xE4;ngige Variable innerhalb einer trigonometrischen Funktion auftritt.<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:25%;"><img style="aspect-ratio:300/24;" src="https://cloudfront.jove.com/files/ftp_upload/19805/19805_Equation_2_editor_19805.svg" alt="19805_Equation_2.svg" /></figure>Die Differentiation beider Seiten nach der unabh&#xE4;ngigen Variable liefert eine Beziehung zwischen dem Kosinus des Winkels und der &#xC4;nderungsrate des Winkels. Durch Umstellen dieser Beziehung l&#xE4;sst sich die Ableitung des Arkussinus isolieren. Der Kosinusterm wird anschlie&#xDF;end unter Verwendung einer trigonometrischen Identit&#xE4;t, die den Kosinus durch den Sinus ausdr&#xFC;ckt, umgeschrieben. Da der Sinus des Winkels bereits aus der urspr&#xFC;nglichen Beziehung bekannt ist, f&#xFC;hrt diese Substitution zum endg&#xFC;ltigen Ausdruck f&#xFC;r die Ableitung.<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:25%;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19805/19805_Equation_3_editor_19805.svg" alt="19805_Equation_3.svg" /></figure>Dieses Ergebnis besitzt eine wichtige physikalische Bedeutung. N&#xE4;hert sich das Flugzeug einer Position direkt &#xFC;ber dem Beobachter, n&#xE4;hert sich das Verh&#xE4;ltnis von H&#xF6;he zu Entfernung seinem Grenzwert. In dieser N&#xE4;he bewirken bereits kleinste &#xC4;nderungen des Verh&#xE4;ltnisses sehr gro&#xDF;e &#xC4;nderungen des H&#xF6;henwinkels. Dieser rapide Anstieg der Empfindlichkeit f&#xFC;hrt zu Instabilit&#xE4;t bei der Nachf&#xFC;hrung und kann letztlich zum Ausfall der Nachf&#xFC;hrung f&#xFC;hren. &#xC4;hnliche Verfahren des impliziten Differenzierens lassen sich anwenden, um die Ableitungen der &#xFC;brigen inversen trigonometrischen Funktionen zu ermitteln, die in der N&#xE4;he ihrer Grenzwerte ein analoges Empfindlichkeitsverhalten zeigen.

Ableitungen inverser trigonometrischer Funktionen

Ein Schiff, das ein sich n&#xE4;herndes Flugzeug verfolgt, nutzt geometrische Messungen, um dessen Position relativ zum Beobachter zu bestimmen. Durch die ...

derivatives-of-inverse-trigonometric-functions

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inverse trig derivatives in tracking

Inverse trigonometric derivatives can describe how a tracking angle changes as an aircraft moves closer to an observer. In this situation, a ship measures the slant distance to the aircraft and the angle of elevation. These measurements let the observer find the horizontal and vertical parts of the distance using trigonometric relationships.

The angle of elevation is treated as the dependent variable, and the ratio of altitude to slant distance is treated as the independent variable. That setup creates an inverse trigonometric relationship. The key idea is tracking sensitivity, which means how strongly a small change in the ratio affects the angle.

To find the derivative, the inverse relation is first rewritten in trig form by applying the sine function to both sides. This step makes implicit differentiation possible, since the angle appears inside a trig function. Differentiating with respect to the independent variable gives a term with cosine of the angle and the rate of change of the angle.

Rearranging the equation isolates the derivative of the inverse sine function. The cosine term is then replaced with a trig identity that writes cosine in terms of sine. Because the sine value is already known from the original relationship, the final derivative can be written in a simpler form.

This result has a clear physical meaning. As the aircraft moves toward a point directly overhead, the ratio of altitude to distance approaches its limit. Near that limit, very small changes in the ratio cause very large changes in the angle of elevation. That sharp rise in sensitivity can make tracking unstable and eventually fail. The same implicit differentiation method also gives the derivatives of the other inverse trigonometric functions, and they show similar limiting behavior near their edges.

Inverse Trig Derivatives in Tracking

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key terms and concepts

Learn how inverse trig derivatives describe tracking sensitivity, from an aircraft’s angle of elevation to the implicit differentiation used to find the formula.

Inverse Trig Derivatives in Tracking | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Inverse Trig Derivatives in Tracking

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