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

Verified step by step guidance

1. Recognize that \( \sin^{-1} \) is the inverse function of \( \sin \), which means \( \sin^{-1}(\sin(x)) = x \) for \( x \) in the range of \( \sin^{-1} \).
2. The range of \( \sin^{-1} \) is \([-\frac{\pi}{2}, \frac{\pi}{2}]\).
3. Determine if \( \frac{5\pi}{6} \) is within the range of \( \sin^{-1} \).
4. Since \( \frac{5\pi}{6} \) is not within \([-\frac{\pi}{2}, \frac{\pi}{2}]\), find an equivalent angle within this range that has the same sine value.
5. Use the identity \( \sin(\pi - x) = \sin(x) \) to find the equivalent angle within the range of \( \sin^{-1} \).

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.