Problem 115 - Algebra and Trigonometry Real Mathematics, Real People

Question

Prove the identity \(\arcsin (-x)=-\arcsin x\)

Step-by-Step Solution

Answer

The identity \(\arcsin (-x)=-\arcsin x\) has been proven, considering the properties and definitions of the arcsin function, and its behaviour with respect to the sign of the input.

Step 1: Recall the definition and properties of the arcsin function

The arcsin function, also known as the inverse sine function, is denoted as \(\arcsin x\) or \(\sin^{-1}x\). It has a range of \([- \pi/2, \pi/2]\) and is an increasing function over this interval.

Step 2: Consider the sign of input x

Since the function \(\arcsin x\) is increasing, that means \(\arcsin -x\) falls to the left of \(\arcsin x\) on the x-axis. With the established range of the arcsin function being \([- \pi/2, \pi/2]\), values less than 0 fall into the interval \([- \pi/2, 0)\). Hence, \(\arcsin -x\) will be a negative value.

Step 3: Relate to the negative value of arcsin x

Since the range of arcsin is symmetric about the origin and \(\arcsin x\) lies in the interval \((-\pi/2, \pi/2)\), \(-\arcsin x\) will simply flip the value and thus it will still exist in the interval \((-\pi/2, \pi/2)\). This makes \(-\arcsin x\) the mirror image of \(\arcsin x\) around x-axis.

Step 4: Establish the connection

From the last two steps, it can be concluded that both \(\arcsin (-x)\) and \(-\arcsin x\) refer to the same quantity: a value in the range of \([- \pi/2, 0)\) which is the mirror image of \(\arcsin x\) along x-axis. Hence, they are equal, i.e., \(\arcsin (-x) = -\arcsin x\).

Key Concepts

Understanding the Arcsin Function

The arcsin function, often denoted as \( \arcsin x \) or \( \sin^{-1}x \), is known as the inverse of the sine function. Unlike the sine function, which has a range from -1 to 1, the arcsin function has a specific range to ensure it remains a function. The primary range of the arcsin function is from \[-\frac{\pi}{2}\] to \[\frac{\pi}{2}\]. This includes negative and positive angles, but restricts them to within a 90-degree range. Additionally, due to its nature, the arcsin function is an increasing function over this interval, meaning as the input increases, the output increases as well. This behavior is essential when considering transformations or reflections over the x-axis. Understanding these characteristics helps in exploring identities involving the arcsin, such as \( \arcsin (-x) = -\arcsin x \). By taking the negative of the input, the property of being an odd function is maintained, showing symmetry around the origin.

Range and Domain of Functions

When considering the range and domain of functions such as the arcsin function, it is crucial to delimit these to avoid ambiguity. The domain of \( \arcsin x \) is the set of all possible inputs and is from -1 to 1. This means \( x \) must fall in this interval to produce a real result. The corresponding range of the arcsin function is then \([-\frac{\pi}{2}, \frac{\pi}{2}]\), ensuring that the results are real numbers and preserve the function's characteristics.
- For inputs \( x \) closer to -1, the output approaches \(-\frac{\pi}{2}\).- For inputs closer to 1, the output approaches \(\frac{\pi}{2}\). By confining inputs to this domain and outputs to this range, one can better understand the mechanics behind trigonometric identities and transformations, ensuring results align with expected mathematical behavior.

Properties of Inverse Functions

Inverse functions are fascinating due to their ability to reverse the effect of the original function. The arcsin function is a perfect example of this, as it reverses the sine operation. With inverse functions, the input becomes the output and vice versa, provided the domain and ranges are properly aligned. This alignment is crucial as it lies at the heart of many trigonometric proofs and identities.
- An important property of all inverse trigonometric functions, including arcsin, is that they show symmetry, often around axes or the origin.- Specifically, the arcsin function is an odd function, which implies that \( \arcsin(-x) = -\arcsin(x) \). This property reflects how negative inputs in the domain lead to negative outputs in the range, mirroring the behavior observed with non-inverted functions. Understanding these properties not only simplifies solving trigonometric equations but also enhances comprehension of how trigonometric systems interrelate, offering deeper insights into their geometric and algebraic interpretations.