In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. sin⁻¹ (sin 5π/6)

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as sin⁻¹(x), are used to find the angle whose sine is x. The output of these functions is restricted to specific ranges to ensure they are single-valued. For sine, the range is typically between -π/2 and π/2, which is crucial when evaluating expressions like sin⁻¹(sin θ).

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. Understanding the unit circle helps in determining the sine values for various angles, including those greater than π/2 or less than -π/2, which is essential for evaluating expressions involving sine and its inverse.

Reference Angles

Reference angles are the acute angles formed by the terminal side of an angle and the x-axis. They are used to simplify the evaluation of trigonometric functions for angles outside the first quadrant. For example, the reference angle for 5π/6 is π/6, which helps in finding the sine value and subsequently aids in evaluating sin⁻¹(sin 5π/6) accurately.