Find the exact value of each real number y if it exists. Do not use a calculator.
y = sin⁻¹ 0

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 a given number.
2. Recognize that \( \sin^{-1}(0) \) asks for the angle \( y \) such that \( \sin(y) = 0 \).
3. Recall the unit circle and the sine function: \( \sin(y) = 0 \) at specific angles.
4. Identify the angles on the unit circle where the sine value is 0, which are \( y = 0 \) and \( y = \pi \) (or \( y = 180^\circ \)).
5. Since the range of \( \sin^{-1} \) is \([-\frac{\pi}{2}, \frac{\pi}{2}]\), the angle \( y \) must be \( 0 \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as sin⁻¹ (arcsin), are used to find the angle whose sine is a given number. For example, if y = sin⁻¹(0), we are looking for an angle θ such that sin(θ) = 0. The range of the arcsin function is limited to [-π/2, π/2], which helps in determining the specific angle.

Sine Function Values

The sine function, which is a fundamental trigonometric function, outputs the ratio of the length of the opposite side to the hypotenuse in a right triangle. The sine of certain angles, such as 0, π/2, and π, is well-known. Specifically, sin(0) = 0, which is crucial for solving the equation y = sin⁻¹(0).

Range and Domain of Trigonometric Functions

Understanding the range and domain of trigonometric functions is essential for solving inverse functions. The sine function has a domain of all real numbers and a range of [-1, 1]. Conversely, the arcsin function has a domain of [-1, 1] and a range of [-π/2, π/2]. This knowledge helps in identifying valid inputs and outputs for the functions involved.