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 th…
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.
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Q1: How do you find the derivative of the arcsine function?
To find the derivative of arcsin(x), rewrite the inverse relationship in equivalent trigonometric form by applying sine to both sides, giving sin(y) = x. Then use implicit differentiation with respect to x. Rearrange the result and apply the Pythagorean identity to express cosine in terms of sine. Since sin(y) = x, substitute to obtain the final derivative formula.
Q2: Why does tracking sensitivity increase as an aircraft approaches directly overhead?
As the aircraft moves closer to directly overhead, the ratio of altitude to slant 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, quantified by the derivative, leads to tracking instability and ultimately causes tracking failure.
Q3: What is the relationship between angle of elevation and the altitude-to-distance ratio?
The angle of elevation is related to the altitude-to-distance ratio through an inverse trigonometric function, expressed as y = arcsin(x), where y is the angle and x is the ratio. This inverse relationship forms the basis for studying tracking sensitivity, which describes how strongly small changes in the ratio affect the measured angle.
Q4: How does implicit differentiation apply to inverse trigonometric functions?
Implicit differentiation allows you to find derivatives of inverse trigonometric relationships without explicitly solving for the dependent variable. By rewriting the inverse relationship in trigonometric form and differentiating both sides with respect to the independent variable, you can isolate the derivative. This technique reveals how the angle changes instantaneously relative to changes in measured quantities.
Q5: What trigonometric identity is used when differentiating arcsin(x)?
The Pythagorean identity is applied to express cosine in terms of sine during the differentiation process. Since sin(y) = x from the original inverse relationship, you can substitute this into the identity to eliminate the cosine term and express the final derivative entirely in terms of the independent variable x.
Q6: How can a ship use geometric measurements to track an approaching aircraft?
A ship measures the slant distance to the aircraft and the angle of elevation. Using trigonometric relationships, these measurements yield the horizontal and vertical components of distance. Analyzing how the angle changes mathematically through derivatives reveals the tracking sensitivity, enabling the ship to understand how position changes affect observed angle measurements.
Q7: Can the same differentiation method be applied to other inverse trigonometric functions?
Yes, similar implicit differentiation techniques can be applied to find derivatives of the remaining inverse trigonometric functions, such as arccos and arctan. Each displays analogous sensitivity behavior near its limits, where small changes in the input produce large changes in the output angle, similar to the arcsin case.