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 \( \sin^{-1} \) is the inverse sine function, also known as arcsine.
2. Understand that \( \sin(\sin^{-1}(x)) = x \) for \( x \) in the range \([-1, 1]\).
3. Since 0.9 is within the range \([-1, 1]\), the expression \( \sin(\sin^{-1}(0.9)) \) simplifies directly.
4. Apply the property of inverse functions: \( \sin(\sin^{-1}(0.9)) = 0.9 \).
5. Conclude that the exact value of the expression is simply 0.9.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as sin⁻¹(x), are used to find the angle whose sine is x. For example, sin⁻¹(0.9) gives the angle θ such that sin(θ) = 0.9. Understanding this concept is crucial for evaluating expressions involving inverse functions.

Sine Function

The sine function is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse. Knowing how to evaluate sin(θ) for specific angles or values is essential for solving trigonometric expressions.

Composition of Functions

Composition of functions involves applying one function to the result of another. In this case, evaluating sin(sin⁻¹(0.9)) means finding the sine of the angle whose sine is 0.9. This concept is important for simplifying expressions and understanding how functions interact.