In Exercises 27–38, use a calculator to find the value of each expression rounded to two decimal places. sin⁻¹ (-0.32)

Verified step by step guidance

1. Understand that \( \sin^{-1} \) is the inverse sine function, also known as arcsin, which gives the angle whose sine is the given number.
2. Recognize that the range of the \( \sin^{-1} \) function is \([-\frac{\pi}{2}, \frac{\pi}{2}]\) or \([-90^\circ, 90^\circ]\).
3. Use a calculator to find \( \sin^{-1}(-0.32) \). Ensure the calculator is set to the correct mode (degrees or radians) based on the context of your problem.
4. Input \(-0.32\) into the calculator and press the inverse sine function button to get the angle.
5. Round the resulting angle to two decimal places as required by the problem.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as sin⁻¹ (arcsine), are used to determine the angle whose sine is a given value. For example, sin⁻¹(x) returns the angle θ such that sin(θ) = x. These functions are essential for solving problems where the angle is unknown, and they have specific ranges to ensure each output is unique.

Domain and Range of Sine Function

The sine function has a domain of all real numbers and a range of [-1, 1]. This means that the input for the sine function can be any real number, but the output will always fall between -1 and 1. Consequently, when using the inverse sine function, the input must also be within this range, which is crucial for determining valid outputs.

Calculator Functions and Rounding

Using a calculator to find inverse trigonometric values involves understanding how to input the function correctly and interpret the output. After obtaining the angle in radians or degrees, rounding to two decimal places is often required for precision in reporting results. Familiarity with calculator settings (degrees vs. radians) is also important to ensure accurate calculations.