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

Verified step by step guidance

1. Recognize that \( \sin^{-1} \) is the inverse sine function, also known as arcsin, which returns the angle whose sine is the given value.
2. Understand that \( \sin(\pi) = 0 \) because the sine of \( \pi \) radians (or 180 degrees) is 0.
3. The expression \( \sin^{-1}(\sin(\pi)) \) simplifies to \( \sin^{-1}(0) \).
4. Recall that the range of \( \sin^{-1} \) is \([-\frac{\pi}{2}, \frac{\pi}{2}]\), meaning it returns angles within this interval.
5. Determine the angle within the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\) whose sine is 0, which is 0.

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 have specific ranges; for sin⁻¹(x), the output is restricted to the interval [-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. The sine of an angle corresponds to the y-coordinate of the point on the unit circle. Knowing how to use the unit circle helps in visualizing and calculating the values of trigonometric functions for various angles.

Periodic Nature of Sine Function

The sine function is periodic with a period of 2π, meaning it repeats its values every 2π radians. This periodicity implies that sin(π) = 0, but when using the inverse sine function, we must consider the principal value. Thus, while sin(π) equals 0, sin⁻¹(0) yields a specific angle within the defined range of the inverse function.