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

Un navire suit un avion approchant en mesurant la distance inclinée et l’angle d’élévation. À partir de ces mesures, les composantes horizontale et verticale de la distance peuvent être trouvées à l’aide de relations trigonométriquesPour analyser mathématiquement le changement d’angle, l’angle thêta est cartographié comme y, et le rapport altitude/distance est mappé comme x, donnant la relation y égal à arcsin x.La mesure de la façon dont les changements instantanés de x affectent y est la sensibilité angulaire, ce qui nécessite une différentiation implicite.Le sinus des deux côtés de l’équation fournit une forme trigonométrique équivalente, exprimée par sinus y égal à x.Cette équation est alors différentiée implicitement par rapport à x pour trouver la dérivée.En réarrangeant et en appliquant l’identité pythagoricienne, la fonction cosinus peut s’écrire en termes de sinus. Puisque le sinus de y est égal à x, cela peut être substitué pour obtenir la dérivée de l’arcsin de x.Lorsque l’avion est au-dessus, un léger changement de x provoque un énorme changement d’angle, entraînant une explosion de sensibilité au suivi et une défaillance du suivi. De même, on peut également trouver les dérivées des fonctions trigonométriques inverses restantes.

dérivées des fonctions trigonométriques inverses

Derivatives of Inverse Trigonometric Functions

Un navire qui suit un a&#xE9;ronef en approche s&#x2019;appuie sur des mesures g&#xE9;om&#xE9;triques afin de d&#xE9;terminer la position de l&#x2019;a&#xE9;ronef par rapport &#xE0; l&#x2019;observateur. En mesurant la distance oblique &#xE0; l&#x2019;a&#xE9;ronef ainsi que l&#x2019;angle d&#x2019;&#xE9;l&#xE9;vation, il est possible de d&#xE9;terminer les composantes horizontale et verticale de cette distance &#xE0; l&#x2019;aide de relations trigonom&#xE9;triques. Cette approche g&#xE9;om&#xE9;trique fournit une base permettant d&#x2019;analyser la mani&#xE8;re dont l&#x2019;angle observ&#xE9; &#xE9;volue &#xE0; mesure que l&#x2019;a&#xE9;ronef se rapproche du navire.Afin d&#x2019;&#xE9;tudier le comportement math&#xE9;matique de l&#x2019;angle d&#x2019;&#xE9;l&#xE9;vation, cet angle est repr&#xE9;sent&#xE9; par une variable d&#xE9;pendante, tandis que le rapport entre l&#x2019;altitude de l&#x2019;a&#xE9;ronef et sa distance oblique est repr&#xE9;sent&#xE9; par une variable ind&#xE9;pendante. Cette mise en correspondance conduit &#xE0; une relation trigonom&#xE9;trique inverse entre les deux variables, laquelle constitue la base de l&#x2019;&#xE9;tude de la sensibilit&#xE9; de suivi. La sensibilit&#xE9; caract&#xE9;rise l&#x2019;ampleur de l&#x2019;effet de faibles variations du rapport sur l&#x2019;angle mesur&#xE9; et est quantifi&#xE9;e par diff&#xE9;rentiation.<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>Pour diff&#xE9;rencier cette relation inverse, on la r&#xE9;&#xE9;crit d&#x2019;abord sous une forme trigonom&#xE9;trique &#xE9;quivalente en appliquant la fonction sinus aux deux membres de l&#x2019;&#xE9;galit&#xE9;. Cette transformation permet d&#x2019;utiliser la d&#xE9;rivation implicite, n&#xE9;cessaire puisque la variable d&#xE9;pendante appara&#xEE;t &#xE0; l&#x2019;int&#xE9;rieur d&#x2019;une fonction trigonom&#xE9;trique.<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>En diff&#xE9;renciant les deux membres par rapport &#xE0; la variable ind&#xE9;pendante, on obtient une relation faisant intervenir le cosinus de l&#x2019;angle ainsi que le taux de variation de celui-ci. En r&#xE9;arrangeant cette expression, on peut isoler la d&#xE9;riv&#xE9;e de la fonction arcsinus. Le terme en cosinus est ensuite r&#xE9;&#xE9;crit &#xE0; l&#x2019;aide d&#x2019;une identit&#xE9; trigonom&#xE9;trique exprimant le cosinus en termes de sinus. &#xC9;tant donn&#xE9; que le sinus de l&#x2019;angle est d&#xE9;j&#xE0; connu d&#x2019;apr&#xE8;s la relation initiale, cette substitution permet d&#x2019;obtenir l&#x2019;expression finale de la d&#xE9;riv&#xE9;e.<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>Ce r&#xE9;sultat admet une interpr&#xE9;tation physique importante. Lorsque l&#x2019;a&#xE9;ronef se rapproche d&#x2019;une position situ&#xE9;e &#xE0; la verticale de l&#x2019;observateur, le rapport entre l&#x2019;altitude et la distance oblique tend vers sa valeur limite. &#xC0; proximit&#xE9; de cette situation, des variations extr&#xEA;mement faibles de ce rapport engendrent des variations tr&#xE8;s importantes de l&#x2019;angle d&#x2019;&#xE9;l&#xE9;vation. Cette augmentation rapide de la sensibilit&#xE9; entra&#xEE;ne une instabilit&#xE9; du suivi et conduit, &#xE0; terme, &#xE0; un &#xE9;chec du suivi. Des techniques de d&#xE9;rivation implicite analogues peuvent &#xEA;tre mises en &#x153;uvre pour d&#xE9;terminer les d&#xE9;riv&#xE9;es des autres fonctions trigonom&#xE9;triques inverses, lesquelles pr&#xE9;sentent un comportement de sensibilit&#xE9; comparable pr&#xE8;s de leurs limites.

Dérivées des fonctions trigonométriques inverses

Un navire qui suit un a&#xE9;ronef en approche s&#x2019;appuie sur des mesures g&#xE9;om&#xE9;triques afin de d&#xE9;terminer la position de ...

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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derivatives of inverse trigonometric functions

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...

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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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