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

Verified step by step guidance

1. Recognize that the expression is \( \sin(\sin^{-1} \pi) \). The function \( \sin^{-1} \) (also called arcsin) is the inverse sine function, which returns an angle whose sine is the given value.
2. Recall the domain and range of the inverse sine function: \( \sin^{-1} x \) is defined only for \( x \) in the interval \( [-1, 1] \), and it returns an angle \( \theta \) in the range \( [-\frac{\pi}{2}, \frac{\pi}{2}] \).
3. Since \( \pi \approx 3.14159 \) is outside the domain \( [-1, 1] \) of \( \sin^{-1} \), the expression \( \sin^{-1} \pi \) is not defined in the real numbers.
4. Therefore, the expression \( \sin(\sin^{-1} \pi) \) cannot be evaluated as a real number because \( \sin^{-1} \pi \) does not exist in the real domain.
5. Conclude that the exact value of \( \sin(\sin^{-1} \pi) \) is undefined or does not exist in the real number system.

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 number. Its output range is limited to [-π/2, π/2], meaning it only accepts inputs between -1 and 1 for real values.

Domain Restrictions of the Sine Function

The sine function outputs values between -1 and 1, so its inverse function sin⁻¹ only accepts inputs within this range. If the input to sin⁻¹ is outside [-1, 1], the expression is undefined in the real number system.

Evaluating Composite Trigonometric Expressions

When evaluating expressions like sin(sin⁻¹ x), if x is within the domain of sin⁻¹, the result simplifies to x. However, if x is outside the domain, the expression cannot be evaluated without extending to complex numbers.