In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. sin⁻¹ (sin 5π/6)

검증된 단계별 안내

Recognize that the expression is \( \sin^{-1}(\sin(\frac{5\pi}{6})) \), which means we want to find the angle whose sine is \( \sin(\frac{5\pi}{6}) \).

Recall that \( \sin(\frac{5\pi}{6}) = \sin(\pi - \frac{\pi}{6}) = \sin(\frac{\pi}{6}) \), so \( \sin(\frac{5\pi}{6}) = \frac{1}{2} \).

Understand that the function \( \sin^{-1}(x) \) (also called arcsin) returns values in the principal range \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \).

Since \( \sin^{-1}(\sin(\frac{5\pi}{6})) = \sin^{-1}(\frac{1}{2}) \), find the angle in the principal range whose sine is \( \frac{1}{2} \).

Identify that the angle in \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \) with sine \( \frac{1}{2} \) is \( \frac{\pi}{6} \), so \( \sin^{-1}(\sin(\frac{5\pi}{6})) = \frac{\pi}{6} \).

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주요 개념

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Inverse Sine Function (sin⁻¹ or arcsin)

The inverse sine function, denoted sin⁻¹ or arcsin, returns the angle whose sine is a given value. Its output is restricted to the principal range of [-π/2, π/2], meaning the result must lie within this interval to be considered the principal value.

Sine Function Periodicity and Symmetry

The sine function is periodic with period 2π and is symmetric about the origin (odd function). This means sin(θ) = sin(π - θ), which helps in finding equivalent angles within the principal range when evaluating inverse sine expressions.

Evaluating sin⁻¹(sin θ) for Angles Outside the Principal Range

When θ is outside the principal range of arcsin, sin⁻¹(sin θ) equals the angle within [-π/2, π/2] that has the same sine value as θ. This often involves using angle identities or symmetry to find the equivalent angle inside the principal range.