The Mystery of ( sin^{-1}(sin x) eq x )

The confusion surrounding the behavior of the function ( sin^{-1}(sin x) ) often arises because of the specific domains and ranges of the sine and inverse sine functions. This article aims to demystify this concept by explaining the foundational properties of these functions and their corresponding inverses.

Domain and Range: Key to Understanding

For any function and its inverse, understanding the domain and range is crucial. Let's start with the sine function, denoted as ( sin x ).

Domain and Range of the Sine Function

The sine function, ( sin x ), can take any real number as its input, making its domain the set of all real numbers ( mathbb{R} ). However, the output or range of ( sin x ) is constrained to the interval ([-1, 1]), as the sine function only produces values between -1 and 1.

Domain and Range of the Inverse Sine Function

The inverse sine function, also known as the arcsine function, denoted as ( sin^{-1} x ) or ( arcsin x ), operates in a different manner. Its domain is restricted to ([-1, 1]), capturing the full range of the sine function. Furthermore, the range of ( sin^{-1} x ) is limited to the interval ([- frac{pi}{2}, frac{pi}{2}]), meaning it only covers angles between (-frac{pi}{2}) and (frac{pi}{2}).

Cancellation Condition: When Does ( sin^{-1}(sin x) ) Equal ( x )?

The equation ( sin^{-1}(sin x) x ) holds true only when ( x ) lies within the specified range of the arcsine function, which is ([- frac{pi}{2}, frac{pi}{2}]). Outside this interval, ( sin^{-1}(sin x) ) does not return ( x ), but rather an equivalent angle within the range ([- frac{pi}{2}, frac{pi}{2}]).

Example: ( x frac{pi}{4} )

Consider ( x frac{pi}{4} ). Since ( frac{pi}{4} ) falls within the valid range of the arcsine function, we have:

$$ sin^{-1}(sin frac{pi}{4}) frac{pi}{4} $$

Example: ( x frac{3pi}{4} )

For ( x frac{3pi}{4} ), which is outside the valid range, we have:

$$ sin^{-1}(sin frac{3pi}{4}) frac{pi}{4} $$

Note that ( frac{3pi}{4} ) is equivalent to ( pi - frac{pi}{4} ), and the arcsine function returns the angle within the range ([- frac{pi}{2}, frac{pi}{2}]) that has the same sine value.

General Case: Understanding the Mappings

For any value of ( x ), the expression ( sin^{-1}(sin x) ) can be written as a piecewise function based on the specific intervals of ( x ):

$$ sin^{-1}(sin x) begin{cases} x text{if } -frac{pi}{2} leq x leq frac{pi}{2} pi - x text{if } frac{pi}{2} x frac{3pi}{2} -2pi x text{if } x geq 2pi -2pi x text{for similar ranges} end{cases} $$

Conclusion: The Intricacies of Inverse Functions

Thus, while ( sin^{-1} x ) and ( sin x ) can effectively cancel each other out, this cancellation only occurs within specific intervals. This principle applies to all trigonometric functions and their inverses, such as ( cos^{-1}(cos x) ) and ( tan^{-1}(tan x) ), which also display similar behavior based on their respective domains and ranges.

Understanding the domain and range of these functions is essential to grasp the concept of inverse functions and how they interact.