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

A ship tracks an approaching aircraft by measuring the slant distance and the angle of elevation. From these measurements, the horizontal and vertical components of the distance can be found using trigonometric relationshipsTo analyze the change in angle mathematically, angle theta is mapped as y, and the Altitude over distance ratio is mapped as x, giving the relation y equals arcsin x.The measure of how instantaneous changes in x affect y is the angle sensitivity, and this requires implicit differentiation.The sine of both sides of the equation provides an equivalent trigonometric form, expressed as sine y equals x.This equation is then differentiated implicitly with respect to x to find the derivative.By rearranging and applying the Pythagorean identity, the cosine function can be written in terms of sine. Since the sine of y equals x, this can be substituted back to get the derivative of arcsin of x.When the aircraft is overhead, a tiny change in x produces a huge change in the angle, resulting in tracking sensitivity explosion and subsequent tracking failure. Similarly, the derivatives of the remaining inverse trigonometric functions can also be found.

derivatives of inverse trigonometric functions

A ship tracking an approaching aircraft relies on geometric measurements to find out the aircraft&#8217;s position relative to the observer. By measuring the slant distance to the aircraft and the angle of elevation, the horizontal and vertical components of the distance can be obtained using trigonometric relationships. This geometric approach provides a basis for analyzing how the observed angle changes as the aircraft moves closer to the ship.To examine the mathematical behavior of the angle of elevation, the angle is represented by a dependent variable, while the ratio of the aircraft&#8217;s altitude to its slant distance is represented by an independent variable. This mapping leads to an inverse trigonometric relationship between the two variables, which forms the basis for studying tracking sensitivity. The sensitivity describes how strongly small changes in the ratio affect the measured angle and is quantified using differentiation.<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:29.94%;"><img style="aspect-ratio:300/24;" src="https://cloudfront.jove.com/files/ftp_upload/19805/19805_Equation_1_editor_19805.svg" alt="y = arcsin x function, mathematical equation, educational diagram, inverse sine concept." data-altupdatedat="1775223722000" data-alt="19805_Equation_1.svg" /></figure>To differentiate this inverse relationship, it is first rewritten in an equivalent trigonometric form by applying the sine function to both sides. This transformation allows the use of implicit differentiation, which is necessary because the dependent variable appears inside a trigonometric function.<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:30.88%;"><img style="aspect-ratio:300/24;" src="https://cloudfront.jove.com/files/ftp_upload/19805/19805_Equation_2_editor_19805.svg" alt="Sine function equation, sin(y)=x, mathematical concept diagram for trigonometry studies." data-altupdatedat="1775223722000" data-alt="19805_Equation_2.svg" /></figure>Differentiating both sides with respect to the independent variable produces a relationship involving the cosine of the angle and the rate of change of the angle. By rearranging this result, the derivative of the inverse sine function can be isolated. The cosine term is then rewritten using a trigonometric identity that expresses cosine in terms of sine. Since the sine of the angle is already known from the original relationship, this substitution leads to the final expression for the derivative.<figure class="image image-style-align-center" style="display:table;margin-left:auto;margin-right:auto;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19805/19805_Equation_3_editor_19805.svg" alt="Differential equation dy/dx=1/&#8730;(1&#8722;x&#178;), calculus concept, mathematical formula." data-altupdatedat="1775223722000" data-alt="19805_Equation_3.svg" /></figure>This result has an important physical interpretation. As the aircraft approaches a position directly overhead, the ratio of altitude to distance approaches its limiting value. Near this condition, extremely small changes in the ratio produce very large changes in the angle of elevation. This rapid increase in sensitivity leads to tracking instability and ultimately results in tracking failure. Similar implicit differentiation techniques can be applied to get the derivatives of the remaining inverse trigonometric functions, which display analogous sensitivity behavior near their limits.

Derivatives of Inverse Trigonometric Functions

A ship tracking an approaching aircraft relies on geometric measurements to find out the aircraft&#8217;s position relative to the observer. By measuring ...

derivatives-of-inverse-trigonometric-functions

Education

JoVE Core

Calculus

Unit

Differentiation Rules

KEY TERMS AND CONCEPTS

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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