Understanding the Difference Between Arctanx, Arcsinx, and Arccosx in Calculus and Trigonometry

In mathematics, particularly in calculus and trigonometry, the functions arctanx, arcsinx, and arccosx play crucial roles in understanding the relationships between angles and their corresponding trigonometric ratios. It is important to clarify that these functions do not yield the same values and are not interchangeable. This article will delve into why arctanx is not equal to arcsinx/arccosx.

Understanding the Components

The ratio of sin x to cos x, which is the tangent function, or tan x, represents a constant measurement of distance ratios. However, arcsin x and arccos x are defined as angles corresponding to given sine and cosine values, not their ratios. This fundamental difference is the key to understanding why arctanx is distinct from arcsinx and arccosx.

Example Exploring arcsin x 45° and arccos x 45°

Consider the case where sin x cos x 1/√2. We know that tan x 1, thus x 45°. However, if we attempt to equate arcsin x / arccos x, we get 45° / 45° 1. This is clearly not equal to arctan x, which would be 45°. This example illustrates the inherent distinction between the three inverse trigonometric functions.

Further Clarification with Formulas

Another related formula is tan x sin x / cos x. However, attempting to equate arctan x arcsin x / arccos x would be irrational. This is because arctan x specifically aims to find the angle x° given the tangent value, while arcsin x and arccos x determine angles based on the sine and cosine values, respectively.

The Reason Behind the Difference

A very simple reason why arctan x, arcsin x, and arccos x are not equal is that they all represent angles measured in units like degrees or radians. The result of arcsin x / arccos x is a number (degrees/degrees), whereas arctan x represents an angle. Therefore, the two cannot be equal.

Mathematical Contradiction

Assuming that arcsin x arctan x, arccos x would yield a contradiction. For instance, if x 1, the right side of the equation would be arcsin 1 π/2. The left side of the equation would be arctan 1 arccos 1 π/4 × 0 0. This contradiction demonstrates the invalidity of the assumption.

Conclusion

The relationship between arctan x, arcsin x, and arccos x is rooted in their definitions and the nature of the trigonometric functions they represent. Understanding these differences is crucial for solving complex problems in calculus and trigonometry. Confusing or equating these functions can lead to incorrect solutions, as illustrated in the examples provided.