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

Statek śledzi nadlatujący samolot, mierząc odległość nachylenia i kąt elevacji. Z tych pomiarów można wyznaczyć poziome i pionowe składowe odległości za pomocą relacji trygonometrycznychAby matematycznie przeanalizować zmianę kąta, kąt theta odwzorowuje się jako y, a stosunek wysokości do odległości jako x, co daje relację y równe arcsin x.Miarą, jak natychmiastowe zmiany x wpływają na y, jest czułość kątowa, co wymaga niejawnej różniczkowania.Sinus obu stron równania zapewnia równoważną formę trygonometryczną, wyrażoną jako sinus y równy się x.Równanie to jest następnie różniczkowane niejawnie względem x, aby znaleźć pochodną.Poprzez przeorganizowanie i zastosowanie tożsamości pitagorejskiej, funkcję cosinus można zapisać za pomocą sinusa. Ponieważ sinus y równa się x, można to podstawić, aby otrzymać pochodną łuku x.Gdy samolot znajduje się nad głową, niewielka zmiana x powoduje ogromną zmianę kąta, skutkując eksplozją czułości śledzenia i późniejszą awarią śledzenia. Podobnie można znaleźć pochodne pozostałych odwrotnych funkcji trygonometrycznych.

pochodne odwrotnych funkcji trygonometrycznych

Derivatives of Inverse Trigonometric Functions

Statek &#x15B;ledz&#x105;cy zbli&#x17C;aj&#x105;cy si&#x119; samolot opiera si&#x119; na pomiarach geometrycznych w celu okre&#x15B;lenia po&#x142;o&#x17C;enia samolotu wzgl&#x119;dem obserwatora. Mierz&#x105;c odleg&#x142;o&#x15B;&#x107; sko&#x15B;n&#x105; do samolotu oraz k&#x105;t wzniesienia, mo&#x17C;na uzyska&#x107; sk&#x142;adowe poziom&#x105; i pionow&#x105; odleg&#x142;o&#x15B;ci za pomoc&#x105; zale&#x17C;no&#x15B;ci trygonometrycznych. To podej&#x15B;cie geometryczne stanowi podstaw&#x119; do analizy tego, jak zmienia si&#x119; obserwowany k&#x105;t, gdy samolot zbli&#x17C;a si&#x119; do statku.Aby zbada&#x107; matematyczne w&#x142;asno&#x15B;ci k&#x105;ta wzniesienia, k&#x105;t przedstawia si&#x119; jako zmienn&#x105; zale&#x17C;n&#x105;, a stosunek wysoko&#x15B;ci samolotu do jego odleg&#x142;o&#x15B;ci sko&#x15B;nej przedstawia si&#x119; jako zmienn&#x105; niezale&#x17C;n&#x105;. To odwzorowanie prowadzi do zale&#x17C;no&#x15B;ci z odwrotn&#x105; funkcj&#x105; trygonometryczn&#x105; mi&#x119;dzy tymi dwiema zmiennymi, kt&#xF3;ra stanowi podstaw&#x119; analizy czu&#x142;o&#x15B;ci &#x15B;ledzenia. Czu&#x142;o&#x15B;&#x107; opisuje, jak silnie niewielkie zmiany w stosunku wp&#x142;ywaj&#x105; na mierzony k&#x105;t i jest kwantyfikowana za pomoc&#x105; r&#xF3;&#x17C;niczkowania.<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>Aby zr&#xF3;&#x17C;niczkowa&#x107; t&#x119; zale&#x17C;no&#x15B;&#x107;, najpierw przekszta&#x142;ca si&#x119; j&#x105; w r&#xF3;wnowa&#x17C;n&#x105; posta&#x107; trygonometryczn&#x105;, stosuj&#x105;c funkcj&#x119; sinus do obu stron. Ta transformacja pozwala na zastosowanie r&#xF3;&#x17C;niczkowania niejawnego, co jest konieczne, poniewa&#x17C; zmienna zale&#x17C;na znajduje si&#x119; wewn&#x105;trz funkcji trygonometrycznej.<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>R&#xF3;&#x17C;niczkowanie obu stron wzgl&#x119;dem zmiennej niezale&#x17C;nej prowadzi do zale&#x17C;no&#x15B;ci obejmuj&#x105;cej cosinus k&#x105;ta oraz tempo zmiany k&#x105;ta. Przekszta&#x142;caj&#x105;c ten wynik, mo&#x17C;na wydzieli&#x107; pochodn&#x105; funkcji arcus sinus. Wyraz z cosinusem jest nast&#x119;pnie zapisywany za pomoc&#x105; to&#x17C;samo&#x15B;ci trygonometrycznej, kt&#xF3;ra wyra&#x17C;a cosinus za pomoc&#x105; sinusa. Poniewa&#x17C; sinus k&#x105;ta jest ju&#x17C; znany z pierwotnej zale&#x17C;no&#x15B;ci, to podstawienie prowadzi do ostatecznego wyra&#x17C;enia na pochodn&#x105;.<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>Ten wynik ma wa&#x17C;n&#x105; interpretacj&#x119; fizyczn&#x105;. Gdy samolot zbli&#x17C;a si&#x119; do pozycji bezpo&#x15B;rednio nad statkiem, stosunek wysoko&#x15B;ci do odleg&#x142;o&#x15B;ci zbli&#x17C;a si&#x119; do warto&#x15B;ci granicznej. W tym stanie skrajnie ma&#x142;e zmiany tego stosunku powoduj&#x105; bardzo du&#x17C;e zmiany k&#x105;ta wzniesienia. Ten gwa&#x142;towny wzrost czu&#x142;o&#x15B;ci prowadzi do niestabilno&#x15B;ci &#x15B;ledzenia i ostatecznie do awarii &#x15B;ledzenia. Podobne techniki niejawnego r&#xF3;&#x17C;niczkowania mo&#x17C;na zastosowa&#x107; w celu uzyskania pochodnych pozosta&#x142;ych odwrotnych funkcji trygonometrycznych, kt&#xF3;re wykazuj&#x105; analogiczne zachowanie czu&#x142;o&#x15B;ci w pobli&#x17C;u swoich granic.

Pochodne odwrotnych funkcji trygonometrycznych

Statek &#x15B;ledz&#x105;cy zbli&#x17C;aj&#x105;cy si&#x119; samolot opiera si&#x119; na pomiarach geometrycznych w celu okre&#x15B;lenia ...

derivatives-of-inverse-trigonometric-functions

Education

JoVE Core

Calculus

Rachunek różniczkowy i całkowy

Unit

Jednostka

Differentiation Rules

Zasady różniczkowania

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

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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