Find the exact value of each real number y. Do not use a calculator.
y = sin⁻¹ √2/2

Verified step by step guidance

1. Recognize that \( y = \sin^{-1} \left( \frac{\sqrt{2}}{2} \right) \) means we are looking for an angle \( y \) such that \( \sin(y) = \frac{\sqrt{2}}{2} \).
2. Recall the unit circle values: \( \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \) and \( \sin(\frac{3\pi}{4}) = \frac{\sqrt{2}}{2} \).
3. Since \( \sin^{-1} \) (also known as arcsin) returns values in the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\), we need to find the angle within this range.
4. Identify that \( \frac{\pi}{4} \) is within the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\) and satisfies \( \sin(y) = \frac{\sqrt{2}}{2} \).
5. Conclude that \( y = \frac{\pi}{4} \) is the exact value of the angle.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as sin⁻¹ (arcsine), are used to find the angle whose sine is a given value. For example, if y = sin⁻¹(x), then sin(y) = x. Understanding how to interpret these functions is crucial for solving problems involving angles and their corresponding trigonometric ratios.

Unit Circle

The unit circle is a fundamental concept in trigonometry that defines the relationship between angles and their sine and cosine values. It is a circle with a radius of one centered at the origin of a coordinate plane. The coordinates of points on the unit circle correspond to the cosine and sine of the angle formed with the positive x-axis, which helps in determining the exact values of trigonometric functions.

Special Angles

Special angles are commonly used angles in trigonometry, such as 0°, 30°, 45°, 60°, and 90°, for which the sine, cosine, and tangent values are well-known. For instance, sin(45°) = √2/2. Recognizing these angles allows for quick calculations and helps in finding exact values without the need for a calculator.