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

ספינה עוקבת אחרי מטוס מתקרב על ידי מדידת מרחק השיפוע וזווית הגובה. ממדידות אלו ניתן למצוא את הרכיבים האופקיים והאנכיים של המרחק באמצעות קשרים טריגונומטרייםכדי לנתח את שינוי הזווית מתמטית, זווית תטא ממופה כ-y, ויחס הגובה למרחק ממופה כ-x, מה שנותן ליחס y שווה לקשת x.המדד לאופן שבו שינויים מיידיים ב-x משפיעים על y הוא רגישות הזווית, וזה דורש הבדלה מרומזת.הסינוס של שני צידי המשוואה מספק צורה טריגונומטרית שקולה, המובעת כסינוס y שווה ל-x.משוואה זו מתבצעת אז באופן משתמע ביחס ל-x כדי למצוא את הנגזרת.על ידי סידור מחדש ויישום הזהות הפיתגוראית, ניתן לכתוב את פונקציית הקוסינוס במונחי סינוס. מכיוון שהסינוס של y שווה ל-x, ניתן להחליף זאת חזרה כדי לקבל את הנגזרת של ארקסין של x.כאשר המטוס נמצא מעל הראש, שינוי קטן ב-x גורם לשינוי עצום בזווית, מה שמוביל לפיצוץ רגישות במעקב ולכישלון במעקב. באופן דומה, ניתן למצוא גם את הנגזרות של הפונקציות הטריגונומטריות ההפוכות שנותרו.

נגזרות של פונקציות טריגונומטריות הפוכות

Derivatives of Inverse Trigonometric Functions

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derivatives-of-inverse-trigonometric-functions

Education

JoVE Core

Calculus

חשבון דיפרנציאלי ואינטגרלי

Unit

יחידה

Differentiation Rules

כללי גזירה

KEY TERMS AND CONCEPTS

מונחי מפתח ומושגים

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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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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