In Exercises 1–26, find the exact value of each expression. _ sin⁻¹ √2/2

Verified step by step guidance

1. Recognize that \( \sin^{-1} \) is the inverse sine function, which means we are looking for an angle whose sine value is \( \frac{\sqrt{2}}{2} \).
2. Recall that the sine of \( \frac{\pi}{4} \) or \( 45^\circ \) is \( \frac{\sqrt{2}}{2} \).
3. Consider the range of the inverse sine function, which is \([-\frac{\pi}{2}, \frac{\pi}{2}]\) or \([-90^\circ, 90^\circ]\).
4. Since \( \frac{\pi}{4} \) is within this range, it is a valid solution for \( \sin^{-1} \left( \frac{\sqrt{2}}{2} \right) \).
5. Conclude that the exact value of the expression is \( \frac{\pi}{4} \) or \( 45^\circ \).

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, where the output is typically restricted to a specific range to ensure it is a function.

Values of Trigonometric Functions

Understanding the exact values of trigonometric functions at key angles (like 0°, 30°, 45°, 60°, and 90°) is essential. For instance, sin(45°) = √2/2, which means that sin⁻¹(√2/2) will yield 45° or π/4 radians, as it is one of the standard angles in trigonometry.

Principal Value Range

The principal value range for the arcsine function is from -π/2 to π/2. This means that when finding sin⁻¹(√2/2), the result must fall within this interval, ensuring that the angle returned is the one that corresponds to the positive sine value of √2/2, which is 45°.