Introduction

In this article, we will explore two fundamental trigonometric identities: (1) sin(sin^{-1} x) x and (2) sin^{-1}(sin x) x. We will analyze these statements to determine under which conditions they are true. Understanding the domain and range of these functions is crucial for correctly applying these identities.

sin(sin^-1 x) x

Letrsquo;s start by examining the first statement sin(sin^{-1} x) x.

Definition of arcsin

The function sin^{-1} x, also known as the arcsine function, is defined for x in the interval [-1, 1]. The arcsine function returns an angle theta such that sin theta x. Therefore, applying the sine function to this angle yields:

sin(sin^{-1} x) x, for -1 leq x leq 1.

This identity is always true for any x belonging to the domain of the arcsin function.

sin^-1(sin x) x

Now letrsquo;s consider the second statement sin^{-1}(sin x) x.

Range of sin x

The sine function sin x is periodic and takes values within the range [-1, 1]. However, the arcsin function sin^{-1} x returns values only within the interval [-frac{pi}{2}, frac{pi}{2}]. This means:

sin^{-1}(sin x) will equal x only if x is within the interval [-frac{pi}{2}, frac{pi}{2}].

For values of x outside this interval, sin^{-1}(sin x) will return an angle within [-frac{pi}{2}, frac{pi}{2}] that has the same sine value as x. Therefore, it will not necessarily equal x.

Conclusion

To summarize, the statement sin(sin^{-1} x) x is always true for any x in the interval [-1, 1].

On the other hand, the statement sin^{-1}(sin x) x is only true for x in the interval [-frac{pi}{2}, frac{pi}{2}].

Thus, the identity that is always true for any x is sin(sin^{-1} x) x, provided that x is within the appropriate domain.

Related Concepts

sin x: Sine is a periodic function that takes values in the range [-1, 1] for all real numbers. arcsin x: Also known as the inverse sine function, it returns an angle in the interval [-frac{pi}{2}, frac{pi}{2}] whose sine is equal to x.

Understanding these concepts will help in solving a wide range of trigonometric problems and in optimizing the use of trigonometric functions in various applications.

Conclusion

By understanding the domain and range of the sine and arcsine functions, we can accurately apply the identities sin(sin^{-1} x) x and sin^{-1}(sin x) x. These identities are fundamental in trigonometry and are widely used in calculus, physics, and engineering.