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. Understand that \( \sin^{-1} \) is the inverse sine function, also known as arcsin, which returns the angle whose sine is the given number.
2. Recognize that the domain of \( \sin^{-1} \) is \([-1, 1]\), meaning it only accepts input values within this range.
3. Note that \( \pi \) is approximately 3.14159, which is outside the domain of \( \sin^{-1} \).
4. Since \( \pi \) is not within the domain of the inverse sine function, \( \sin^{-1} \pi \) is undefined.
5. Conclude that \( \sin(\sin^{-1} \pi) \) cannot be evaluated because \( \sin^{-1} \pi \) is not a valid expression.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as sin⁻¹ (arcsin), are used to find the angle whose sine is a given value. For example, sin⁻¹(x) returns an angle θ such that sin(θ) = x, where the output is restricted to a specific range to ensure it is a function. Understanding this concept is crucial for evaluating expressions involving inverse functions.

Domain and Range of Sine Function

The sine function has a domain of all real numbers and a range of [-1, 1]. This means that the sine of any angle will always yield a value between -1 and 1. When dealing with inverse functions, it is important to recognize that the input to sin⁻¹ must be within this range, which affects the evaluation of expressions like sin(sin⁻¹(π)).

Exact Values of Trigonometric Functions

Exact values of trigonometric functions refer to specific angles where the sine, cosine, and tangent values can be expressed as simple fractions or radicals. For instance, common angles like 0, π/6, π/4, and π/3 have known sine values. In this context, recognizing that π is outside the range of the sine function is essential for determining the validity of the expression sin(sin⁻¹(π)).