Find the exact value of each expression. sin⁻¹ √2/2

검증된 단계별 안내

Recognize that the expression involves the inverse sine function, written as \(\sin^{-1}\), which means we are looking for an angle \(\theta\) such that \(\sin(\theta) = \frac{\sqrt{2}}{2}\).

Recall the range of the inverse sine function \(\sin^{-1}(x)\), which is \([-\frac{\pi}{2}, \frac{\pi}{2}]\) or \([-90^\circ, 90^\circ]\), so the angle we find must lie within this interval.

Identify the common angles where sine values are known, especially those involving \(\frac{\sqrt{2}}{2}\). For example, \(\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\).

Since \(\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\) and \(\frac{\pi}{4}\) is within the range of \(\sin^{-1}\), conclude that \(\sin^{-1}(\frac{\sqrt{2}}{2}) = \frac{\pi}{4}\).

Express the final answer as the exact angle in radians or degrees, depending on the context of the problem.

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주요 개념

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Inverse Sine Function (sin⁻¹ or arcsin)

The inverse sine function, denoted as sin⁻¹ or arcsin, returns the angle whose sine value is a given number. It is defined for inputs between -1 and 1 and outputs angles typically in the range [-π/2, π/2] or [-90°, 90°]. Understanding this helps find the angle corresponding to a specific sine value.

Exact Values of Sine for Special Angles

Certain angles have well-known sine values expressed in exact terms, such as √2/2 for 45° (π/4 radians). Recognizing these special angles allows you to determine the exact angle without a calculator, which is essential for solving inverse trigonometric expressions exactly.

Domain and Range Restrictions of Inverse Trigonometric Functions

Inverse trigonometric functions have restricted domains and ranges to ensure they are functions. For sin⁻¹, the input must be between -1 and 1, and the output angle lies between -90° and 90°. This restriction ensures a unique solution when finding the angle from a sine value.