In Exercises 29–51, find the exact value of each expression. Do not use a calculator. sin⁻¹(sin π/3)

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as sin⁻¹(x), are used to find the angle whose sine is x. These functions return values within a specific range, which for sin⁻¹(x) is typically [-π/2, π/2]. Understanding this range is crucial for determining the correct angle when evaluating expressions involving inverse sine.

Unit Circle

The unit circle is a fundamental concept in trigonometry that defines the sine and cosine of angles based on their coordinates on a circle with a radius of one. For example, the angle π/3 corresponds to the coordinates (1/2, √3/2), where the sine value is √3/2. Familiarity with the unit circle helps in visualizing and calculating trigonometric values.

Principal Value

The principal value of an inverse trigonometric function is the unique angle that the function returns, constrained to its defined range. For sin⁻¹(sin θ), if θ is outside the range of [-π/2, π/2], the function will return an equivalent angle within this range. This concept is essential for accurately finding the exact value of expressions involving inverse trigonometric functions.