Problem 15 - A Graphical Approach to Precalculus with Limits

Question

Find the exact value of each real number \(y .\) Do not use a calculator. $$y=\sin ^{-1}(-1)$$

Step-by-Step Solution

Answer

The exact value of \( y \) is \(-\frac{\pi}{2}\).

Step 1: Identifying the Problem

We are tasked with finding the value of the inverse sine function, denoted as \(\sin^{-1}(-1)\). This asks us to find an angle \( y \) such that \( \sin(y) = -1 \).

Step 2: Understanding the Range of Inverse Sine

The range of the inverse sine function, \( \sin^{-1}(x) \), is \( [-\frac{\pi}{2}, \frac{\pi}{2}] \). This means the angle solutions for \( y \) must fall within this interval.

Step 3: Solving the Equation

We know \( \sin(y) = -1 \) and we need to find within the inverse sine's range. The sine function equals \(-1\) specifically at \( y = -\frac{\pi}{2} \) in this range.

Step 4: Providing the Calculated Solution

Following the steps, we conclude that if \( \sin(y) = -1 \), within the specific range of \([-\frac{\pi}{2}, \frac{\pi}{2}]\), the real value for \( y \) is \(-\frac{\pi}{2}\).

Key Concepts

Inverse Sine Function

The inverse sine function is a fundamental concept in trigonometry, often symbolized as \( \sin^{-1}(x) \) or \( \arcsin(x) \). When we talk about the inverse sine function, we are interested in finding an angle whose sine is a given value. This setting is valuable in various applications such as solving triangles in geometry, physics, and engineering problems.

• Purpose: Provides the angle associated with a specific sine value.
• Symbol: Represented by \( \sin^{-1}(x) \) or \( \arcsin(x) \).
• Unique Output: Yields a single angle result within a set range.

It's important to differentiate between the inverse sine function and the reciprocal of sine, \( \csc(x) \) or cosecant. This misunderstanding can happen, but remember: the inverse finds angles; reciprocals find ratios.

Angle Solutions

Finding angle solutions is the core task when dealing with inverse trigonometric functions. Understanding which angle corresponds to a trigonometric value is key. To solve for an angle using \( \sin^{-1}(-1) \), imagine searching for an angle \( y \) such that \( \sin(y) = -1 \).

• The equation \( \sin(y) = -1 \) is unique within the specific range of the inverse sine function.
• Within the range \( [-\frac{\pi}{2}, \frac{\pi}{2}] \), this particular sine value ( -1) only occurs at a single point: \( y = -\frac{\pi}{2} \).
• This ensures our solution is exact and precise without the need for calculators.

In practice, finding angle solutions involves analyzing which values fulfill the trigonometric equation within a given set of constraints, ensuring our answers reflect the known properties of the trigonometric function used.

Sine Function Range

Understanding the range of the sine function is crucial when applying its inverse. The range defines possible output values and ensures that inverse trigonometric functions remain one-to-one, essential for providing unique solutions. The range of the sine function itself is between -1 and 1. Conversely, the range of the inverse sine, \( \sin^{-1}(x) \), is \([-\frac{\pi}{2}, \frac{\pi}{2}]\), a crucial piece of information when finding corresponding angles for specific sine values.Here's why the range matters:

• Ensures the inverse sine function is well-defined by providing continuity and predictability for any \( x \) within \([-1, 1]\).
• Guides solutions to fall within a manageable angle set; non-periodic within the range.
• Aids in avoiding ambiguities when seeking precise solutions without calculators.

This predictable range makes it straightforward to identify exact angle solutions like \(-\frac{\pi}{2}\) for \( \sin(y) = -1 \), facilitating deeper understanding and application in various mathematical scenarios.