Find the exact value of each real number y if it exists. Do not use a calculator.
y = sin⁻¹ √3

Key Concepts

Inverse Sine Function

The inverse sine function, denoted as sin⁻¹ or arcsin, is used to find the angle whose sine is a given number. Its range is restricted to [-π/2, π/2] to ensure that it is a function, meaning each input corresponds to exactly one output. Understanding this function is crucial for solving problems involving angles and their sine values.

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 sine of any angle will always yield a value between -1 and 1. When dealing with the inverse sine function, it is essential to recognize that the input must fall within this range; otherwise, the inverse will not yield a real number.

Understanding Square Roots

The square root function, denoted as √, returns the non-negative value that, when squared, equals the input number. In the context of the question, √3 is approximately 1.732, which exceeds the range of the sine function. Therefore, since the sine function cannot produce a value greater than 1, the expression sin⁻¹(√3) does not yield a real number.