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

Un barco rastrea un avión que se acerca midiendo la distancia inclinada y el ángulo de elevación. A partir de estas mediciones, se pueden encontrar los componentes horizontal y vertical de la distancia utilizando relaciones trigonométricasPara analizar matemáticamente el cambio de ángulo, el ángulo theta se mapea como y, y la relación altitud sobre distancia se mapea como x, dando la relación y igual a arcsin x.La medida de cómo los cambios instantáneos en x afectan a y es la sensibilidad angular, y esto requiere una diferenciación implícita.El seno de ambos lados de la ecuación proporciona una forma trigonométrica equivalente, expresada como seno y igual a x.Esta ecuación se diferencia implícitamente respecto a x para encontrar la derivada.Reorganizando y aplicando la identidad pitagórica, la función coseno puede escribirse en términos de seno. Dado que el seno de y es igual a x, esto puede sustituirse para obtener la derivada del arco seno de x.Cuando la aeronave está sobrevolando, un pequeño cambio en x produce un gran cambio en el ángulo, lo que provoca una explosión de sensibilidad al seguimiento y una posterior falla en el seguimiento. De manera similar, también se pueden encontrar las derivadas de las funciones trigonométricas inversas restantes.

derivadas de funciones trigonométricas inversas

Derivatives of Inverse Trigonometric Functions

Una nave que sigue una aeronave que se aproxima se basa en mediciones geom&#xE9;tricas para determinar la posici&#xF3;n de la aeronave con respecto al observador. Al medir la distancia oblicua a la aeronave y el &#xE1;ngulo de elevaci&#xF3;n, se pueden obtener las componentes horizontal y vertical de la distancia mediante relaciones trigonom&#xE9;tricas. Este enfoque geom&#xE9;trico proporciona una base para analizar c&#xF3;mo cambia el &#xE1;ngulo observado a medida que la aeronave se acerca a la nave.Para examinar el comportamiento matem&#xE1;tico del &#xE1;ngulo de elevaci&#xF3;n, este se representa mediante una variable dependiente, mientras que la raz&#xF3;n entre la altitud de la aeronave y su distancia oblicua se representa mediante una variable independiente. Esta correspondencia genera una relaci&#xF3;n trigonom&#xE9;trica inversa entre ambas variables, que constituye la base para estudiar la sensibilidad del rastreo. La sensibilidad describe la magnitud con la que peque&#xF1;os cambios en la raz&#xF3;n afectan el &#xE1;ngulo medido y se cuantifica mediante la diferenciaci&#xF3;n.<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>Para diferenciar esta relaci&#xF3;n inversa, primero se reescribe en una forma trigonom&#xE9;trica equivalente aplicando la funci&#xF3;n seno a ambos lados. Esta transformaci&#xF3;n permite el uso de la diferenciaci&#xF3;n impl&#xED;cita, necesaria porque la variable dependiente aparece dentro de una funci&#xF3;n trigonom&#xE9;trica.<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>Al diferenciar ambos lados con respecto a la variable independiente se obtiene una relaci&#xF3;n que involucra el coseno del &#xE1;ngulo y la derivada del &#xE1;ngulo. Al reordenar este resultado, se puede aislar la derivada de la funci&#xF3;n arcoseno. El t&#xE9;rmino coseno se reescribe entonces utilizando una identidad trigonom&#xE9;trica que expresa el coseno en t&#xE9;rminos del seno. Dado que el seno del &#xE1;ngulo ya se conoce a partir de la relaci&#xF3;n original, esta sustituci&#xF3;n conduce a la expresi&#xF3;n final de la derivada.<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>Este resultado tiene una interpretaci&#xF3;n f&#xED;sica importante. A medida que la aeronave se aproxima a una posici&#xF3;n directamente sobre la vertical del observador, la relaci&#xF3;n entre la altitud y la distancia se aproxima a su valor l&#xED;mite. Cerca de esta situaci&#xF3;n, cambios extremadamente peque&#xF1;os en la relaci&#xF3;n producen cambios muy grandes en el &#xE1;ngulo de elevaci&#xF3;n. Este r&#xE1;pido aumento de la sensibilidad provoca inestabilidad en el rastreo y, en &#xFA;ltima instancia, un fallo del rastreo. Se pueden aplicar t&#xE9;cnicas de diferenciaci&#xF3;n impl&#xED;cita similares para obtener las derivadas de las funciones trigonom&#xE9;tricas inversas restantes, que muestran un comportamiento de sensibilidad an&#xE1;logo cerca de sus l&#xED;mites.

Derivadas de funciones trigonométricas inversas

Una nave que sigue una aeronave que se aproxima se basa en mediciones geom&#xE9;tricas para determinar la posici&#xF3;n de la aeronave con respecto al ...

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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