In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. sin(sin⁻¹ 0.9)

Verified step by step guidance

1. Recognize that the expression is \( \sin(\sin^{-1}(0.9)) \). The function \( \sin^{-1} \) (also called arcsin) is the inverse of the sine function, which means it 'undoes' the sine function within its domain.
2. Recall the property of inverse functions: for any value \( x \) in the domain of \( \sin^{-1} \), \( \sin(\sin^{-1}(x)) = x \). This holds true because \( \sin^{-1}(x) \) gives the angle whose sine is \( x \).
3. Check that the input value \( 0.9 \) is within the domain of \( \sin^{-1} \), which is \( [-1, 1] \). Since \( 0.9 \) is within this range, the property applies directly.
4. Therefore, the expression simplifies directly to \( 0.9 \) without further calculation.
5. Summarize that \( \sin(\sin^{-1}(0.9)) = 0.9 \) because the sine and arcsine functions are inverses on the domain \( [-1, 1] \).

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. It is defined for inputs between -1 and 1 and outputs angles in the range [-π/2, π/2]. Understanding this helps in interpreting expressions like sin(sin⁻¹ x).

Composition of Sine and Inverse Sine Functions

When sine and its inverse are composed as sin(sin⁻¹ x), the result simplifies to x for all x in the domain [-1, 1]. This property is crucial for evaluating expressions without a calculator, as it allows direct simplification.

Domain and Range Restrictions

The domain of sin⁻¹ is [-1, 1], and its range is limited to angles between -π/2 and π/2. Recognizing these restrictions ensures the expression is valid and helps avoid errors when simplifying or evaluating trigonometric expressions.