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Chapter 8: Problem 10

Multiple Choice \(\sin ^{-1}(\sin x)=x\) for all numbers \(x\) for which (a) \(-\infty

Short Answer

Expert verified
The correct answer is (d) \(-\frac{\textbf{蟺}}{2} \leq x \leq \frac{\textbf{蟺}}{2}\).

Step by step solution

01

Understand the Inverse Sine Function

The function \(\text{sin}^{-1}(y)\) is the inverse of \(\text{sin}(x)\). It returns a value \(-\frac{\textbf{蟺}}{2} \leq y \leq \frac{\textbf{蟺}}{2}\). For \(\text{sin}^{-1}(\text{sin} x) = x\) to hold, \(\text{sin} x \) must be in the principal range.
02

Identify the Principal Range of the Inverse Sine Function

To ensure that \(\text{sin}^{-1}(\text{sin} x)\) returns \ x \, \ x \ must lie within the principal range of the inverse sine function. Therefore, \(-\frac{\textbf{蟺}}{2} \leq x \leq \frac{\textbf{蟺}}{2}\).
03

Compare Given Ranges to the Principal Range

Compare the given ranges of \ x \ with \(-\frac{\textbf{蟺}}{2} \leq x \leq \frac{\textbf{蟺}}{2}\): (a) \(-\text{鈭瀩 < x < \text{鈭瀩\) does not match. (b) \(0 \leq x \leq \textbf{蟺}\) does not match. (c) \(-1 \leq x \leq 1\) does not match. (d) \(-\frac{\textbf{蟺}}{2} \leq x \leq \frac{\textbf{蟺}}{2}\) matches.
04

Select the Correct Answer

Based on the comparison, the correct answer is option (d).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Principal Range
The concept of the principal range is crucial for understanding inverse trigonometric functions. When dealing with the inverse sine function, its output values, or the principal range, are restricted to \(-\frac{蟺}{2} \leq y \leq \frac{蟺}{2}\). This limited range ensures that the inverse function returns a unique value for each input. When solving problems involving \(\text{sin}^{-1}\), always make sure that the values lie within this principal range to guarantee a unique and correct answer.
Trigonometric Functions
Trigonometric functions, like sine, cosine, and tangent, relate angles to ratios of sides in right triangles. The sine function, for instance, is defined as the ratio of the length of the side opposite an angle to the hypotenuse. Thus, \(\text{sin}(x) = \frac{\text{opposite}}{\text{hypotenuse}}\). These functions are periodic, meaning they repeat their values over regular intervals. For sine, this interval is 2蟺. Recognizing these properties helps in understanding more complex applications, such as inverses.
Inverse Functions
Inverse functions reverse the roles of input and output. In the context of trigonometry, the inverse sine function \(\text{sin}^{-1}(y)\) undoes the sine function. If \(\text{sin}(x) = y\), then \(\text{sin}^{-1}(y) = x\). The inverse sine function outputs angles within the principal range \(-\frac{蟺}{2} \leq y \leq \frac{蟺}{2}\). This specific range ensures the function is one-to-one, meaning each input corresponds to one unique output.
Sine Function
The sine function is essential in trigonometry and is expressed as \(\text{sin}(x)\). Its values range from -1 to 1 for all real numbers. When dealing with angles, the sine function creates a smooth wave-like pattern, repeating every 2蟺 radians. It's critical to understand both its range and behavior, as these properties underpin many solutions involving trigonometric identities and equations. In problems like \(\text{sin}^{-1}(\text{sin} x) = x\), the principal range and periodic nature of sine play significant roles in determining valid solutions.

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