2.14: Derivatives of Inverse Trigonometric Functions

A ship tracking an approaching aircraft relies on geometric measurements to find out the aircraft’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’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.

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.

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.

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.

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 relationships

To 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.