Problem 210 - Calculus Volume 1

Question

Evaluate the functions. Give the exact value. \(\sin ^{-1}(-1)\)

Step-by-Step Solution

Answer

\(-\frac{\pi}{2}\)

Step 1: Understanding Inverse Sine

The function \( ext{sin}^{-1}(x)\), also called arcsine, is the inverse of the sine function. It returns the angle \(\theta\) such that \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\) and \(\sin(\theta) = x\).

Step 2: Setting Up the Problem

Given \(\sin^{-1}(-1)\), we need to find the angle \(\theta\) for which \(\sin(\theta) = -1\). The angle must be within the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\).

Step 3: Finding the Angle

The angle within the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\) that satisfies \(\sin(\theta) = -1\) is \(\theta = -\frac{\pi}{2}\). At this angle, the sine value is exactly -1, which matches the given condition.

Key Concepts

Arcsine

Understanding the arcsine function is pivotal when dealing with inverse trigonometric functions. The arcsine, denoted as \( \sin^{-1}(x) \) or \( \text{asin}(x) \), is the inverse function of sine. When you use the arcsine function, you are finding the angle \( \theta \) whose sine value is \( x \). This might seem a bit abstract, so think of it this way:

• The arcsine helps us reverse the sine function, which means if \( \sin(\theta) = x \), then \( \theta = \sin^{-1}(x) \).
• The outcome of the arcsine function is an angle.

In the example given in the original exercise, we determined that the angle \( \theta = -\frac{\pi}{2} \) because it is the only angle within the allowable range that has a sine of -1.

Angle

Angles are fundamental in understanding inverse trigonometric functions like arcsine. When we seek the angle for which \( \sin(\theta) = x \), the arcsine function provides this angle. Every angle in trigonometry is measured in radians or degrees, but radians are often preferred in advanced mathematics. Here are some key points about angles in this context:

• The angle resulting from \( \sin^{-1}(x) \) is limited to a specific range, ensuring a unique solution.
• Common angles in radians include \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), \( \frac{\pi}{3} \), \( \frac{\pi}{2} \), and their negatives or equivalents.

Consequently, to find the exact value of \( \sin^{-1}(-1) \), we pinpoint the specific angle \( -\frac{\pi}{2} \) within the defined range of the arcsine function.

Trigonometric Range

When using trigonometric functions, it's crucial to consider their specific ranges. For inverse trigonometric functions such as arcsine, the output is always within a particular range. The range is the set of all possible outcomes (angles in this case). This ensures that, when solving for an angle, the solution is both accurate and unique. Here's what you need to know:

• For \( \sin^{-1}(x) \), the range is \( [-\frac{\pi}{2}, \frac{\pi}{2}] \).
• This range is crucial because it determines where the resulting angle must lie.

This range is important because without it, the solution wouldn't be unique. In our original exercise, this concept guides us to find that \( \sin^{-1}(-1) \) yields \( \theta = -\frac{\pi}{2} \), the only angle within the range where \( \sin(\theta) = -1 \).