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

Verified step by step guidance

1. Recall that the function \(\sin^{-1}(x)\), also known as arcsine, is the inverse of the sine function but its output (range) is restricted to \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\).
2. Identify the input to the arcsine function: here it is \(\sin \pi\). Since \(\sin \pi = 0\), rewrite the expression as \(\sin^{-1}(0)\).
3. Now, find the angle \(\theta\) within the range \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) such that \(\sin \theta = 0\).
4. Recall that \(\sin \theta = 0\) at \(\theta = 0\), \(\pi\), \(2\pi\), etc., but only \(\theta = 0\) lies within the principal range of arcsine.
5. Therefore, the exact value of \(\sin^{-1}(\sin \pi)\) is the angle \(\theta = 0\).

Key Concepts

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] to ensure it is a proper function. Understanding this range is crucial when evaluating expressions like sin⁻¹(sin θ).

Sine Function Periodicity and Symmetry

The sine function is periodic with period 2π, meaning sin(θ) = sin(θ + 2πk) for any integer k. It is also symmetric about the origin (odd function). These properties help simplify expressions and find equivalent angles within the principal range of the inverse sine.

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 finding a reference angle or using angle identities to map θ back into the principal range.