Problem 1 - Precalculus: A Concise Course

Question

Fill in the blanks. $$ \begin{array}{ll} \text { Function } & \text { Alternative Notation } & \text { Domain } & \text { Range } \end{array} $$ $$ y=\arcsin x \quad-\frac{\pi}{2} \leq y \leq \frac{\pi}{2} $$

Step-by-Step Solution

Answer

The alternative notation for the function is \(sin^{-1}(x)\), the domain is \(-1 \leq x \leq 1\), and the range is \(-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}\).

Step 1: Identify Alternate Notation

The alternate notation for the arcsin function is simply \(sin^{-1}(x)\). So, the alternate notation for the function \(y=\arcsin x\) is \(y=sin^{-1}(x)\).

Step 2: Identify Domain of the Function

The domain of the function \(y=\arcsin x\) are the x-values for which the function is defined. Since the arcsin function accepts all real values between -1 and 1, the domain of \(y=\arcsin x\) or equivalently \(y=sin^{-1}(x)\) is \(-1 \leq x \leq 1\).

Step 3: Identify Range of the Function

The range of the function \(y=\arcsin x\) are the returned y-values. It is given that \(-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}\). Thus, the function is defined (i.e., returns real numbers) for all y-values from \(-\frac{\pi}{2}\) up to and including \(\frac{\pi}{2}\). Therefore, the range of the function is \(-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}\).

Key Concepts

Arcsine Function

The arcsine function is one of the foundational inverse trigonometric functions. When we talk about arcsine, represented as \( \arcsin(x) \), we're referring to the inverse of the sine function. In essence, this function helps find the angle whose sine is \( x \). If you know the sine of an angle, the arcsine function will help you determine what that angle is.
The notation \( \arcsin(x) \) is often seen as \( \sin^{-1}(x) \), pronounced "sine inverse." This notation can sometimes be confusing because it looks like an exponent. However, in this context, it specifically denotes the inverse function and not a reciprocal.

• Purpose: Finding an angle with a given sine value.
• Notation: \( \arcsin(x) \) or \( \sin^{-1}(x) \).

Understanding the arcsine function is crucial in trigonometry because it allows us to reverse the sine operation, helping solve equations and tackle problems involving trigonometric relationships.

Domain and Range

When discussing functions, the terms "domain" and "range" are essential. For the arcsine function \( y = \arcsin(x) \), the domain is the set of all possible input values (\( x \) values) for which the function is defined. Here, the domain is limited between \(-1\) and \(1\), meaning \( -1 \leq x \leq 1 \). This restriction comes because the sine function only takes values in this range and the arcsine function is its inverse.
The range of a function is the set of possible output values (\( y \) values) that the function can provide. For arcsine, the range is the intervals of angles in radians, which range from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \).

• Domain: \( -1 \leq x \leq 1 \)
• Range: \( -\frac{\pi}{2} \leq y \leq \frac{\pi}{2} \)

Grasping the concepts of domain and range helps in understanding not just where the function is applicable but also the limitations of its use.

Inverse Trigonometric Functions

Inverse trigonometric functions are a set of functions that reverse the trigonometric functions: sine, cosine, and tangent. These functions are significant because they allow us to find angle measures from known trigonometric ratios. The inverse functions are written with an "arc" prefix, like arcsine \( \arcsin(x) \), or as the function raised to the power of -1, such as \( \sin^{-1}(x) \).
These functions are particularly useful in solving triangles and in many calculus problems, where it is necessary to find an angle with given trigonometric values. Each inverse trigonometric function, including arcsine, has specific domain and range restrictions. These restrictions are essential to ensure that each function has a unique output for every input.

• Inverse Functions: \( \arcsin(x) \), \( \arccos(x) \), \( \arctan(x) \), etc.
• Applications: Used in geometry, trigonometric equations, and calculus problems.

By understanding inverse trigonometric functions, students can confidently move between angles and their trigonometric values, making them a powerful tool in mathematics.