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

Um navio acompanha uma aeronave que se aproxima medindo a distância inclinada e o ângulo de elevação. A partir dessas medições, os componentes horizontal e vertical da distância podem ser encontrados usando relações trigonométricasPara analisar matematicamente a mudança de ângulo, o ângulo theta é mapeado como y, e a razão altitude-distância é mapeada como x, dando a relação y igual a arcsin x.A medida de como as mudanças instantâneas em x afetam y é a sensibilidade angular, e isso requer diferenciação implícita.O seno de ambos os lados da equação fornece uma forma trigonométrica equivalente, expressa como seno y igual a x.Essa equação é então diferenciada implicitamente em relação a x para encontrar a derivada.Ao rearranjar e aplicar a identidade pitagórica, a função cosseno pode ser escrita em termos de seno. Como o seno de y é igual a x, isso pode ser substituído de volta para obter a derivada do arco seno de x.Quando a aeronave está sobrevoando, uma pequena mudança em x produz uma grande alteração no ângulo, resultando em explosão de sensibilidade de rastreamento e subsequente falha no rastreamento. Da mesma forma, as derivadas das funções trigonométricas inversas restantes também podem ser encontradas.

derivadas de funções trigonométricas inversas

Derivatives of Inverse Trigonometric Functions

Um navio que acompanha uma aeronave em aproxima&#xE7;&#xE3;o depende de medi&#xE7;&#xF5;es geom&#xE9;tricas para determinar a posi&#xE7;&#xE3;o da aeronave em rela&#xE7;&#xE3;o ao observador. Medindo a dist&#xE2;ncia obl&#xED;qua at&#xE9; a aeronave e o &#xE2;ngulo de eleva&#xE7;&#xE3;o, as componentes horizontal e vertical da dist&#xE2;ncia podem ser obtidas por meio de rela&#xE7;&#xF5;es trigonom&#xE9;tricas. Essa abordagem geom&#xE9;trica fornece a base para analisar como o &#xE2;ngulo observado muda &#xE0; medida que a aeronave se aproxima do navio.Para examinar o comportamento matem&#xE1;tico do &#xE2;ngulo de eleva&#xE7;&#xE3;o, o &#xE2;ngulo &#xE9; representado por uma vari&#xE1;vel dependente, enquanto a raz&#xE3;o entre a altitude da aeronave e a sua dist&#xE2;ncia obl&#xED;qua &#xE9; representada por uma vari&#xE1;vel independente. Esse mapeamento leva a uma rela&#xE7;&#xE3;o trigonom&#xE9;trica inversa entre as duas vari&#xE1;veis, o que constitui a base para o estudo da sensibilidade de rastreamento. A sensibilidade descreve a intensidade com que pequenas mudan&#xE7;as na raz&#xE3;o afetam o &#xE2;ngulo medido e &#xE9; quantificada por deriva&#xE7;&#xE3;o.<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 derivar essa rela&#xE7;&#xE3;o inversa, ela &#xE9; inicialmente reescrita em uma forma trigonom&#xE9;trica equivalente, aplicando a fun&#xE7;&#xE3;o seno a ambos os lados. Essa transforma&#xE7;&#xE3;o permite o uso da deriva&#xE7;&#xE3;o impl&#xED;cita, necess&#xE1;ria, pois a vari&#xE1;vel dependente aparece dentro de uma fun&#xE7;&#xE3;o 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>Derivando ambos os lados em rela&#xE7;&#xE3;o &#xE0; vari&#xE1;vel independente, obt&#xE9;m-se uma rela&#xE7;&#xE3;o envolvendo o cosseno do &#xE2;ngulo e a taxa de varia&#xE7;&#xE3;o do &#xE2;ngulo. Reorganizando esse resultado, a derivada da fun&#xE7;&#xE3;o arcoseno pode ser isolada. O termo cosseno &#xE9;, ent&#xE3;o, reescrito utilizando uma identidade trigonom&#xE9;trica que expressa o cosseno em termos de seno. Como o seno do &#xE2;ngulo j&#xE1; &#xE9; conhecido a partir da rela&#xE7;&#xE3;o original, essa substitui&#xE7;&#xE3;o leva &#xE0; express&#xE3;o final da derivada.<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:50%;"><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>Esse resultado tem uma importante interpreta&#xE7;&#xE3;o f&#xED;sica. &#xC0; medida que a aeronave se aproxima de uma posi&#xE7;&#xE3;o diretamente acima, a raz&#xE3;o entre altitude e dist&#xE2;ncia se aproxima de seu valor limite. Pr&#xF3;ximo a essa condi&#xE7;&#xE3;o, mudan&#xE7;as extremamente pequenas na raz&#xE3;o produzem altera&#xE7;&#xF5;es muito grandes no &#xE2;ngulo de eleva&#xE7;&#xE3;o. Esse r&#xE1;pido aumento na sensibilidade leva &#xE0; instabilidade de rastreamento e, em &#xFA;ltima inst&#xE2;ncia, resulta em falha no rastreamento. T&#xE9;cnicas semelhantes de deriva&#xE7;&#xE3;o impl&#xED;cita podem ser aplicadas para obter as derivadas das fun&#xE7;&#xF5;es trigonom&#xE9;tricas inversas restantes, que exibem comportamento de sensibilidade an&#xE1;logo pr&#xF3;ximo aos seus limites.

Derivadas de funções trigonométricas inversas

Um navio que acompanha uma aeronave em aproxima&#xE7;&#xE3;o depende de medi&#xE7;&#xF5;es geom&#xE9;tricas para determinar a posi&#xE7;&#xE3;o da ...

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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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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