In Exercises 1–26, find the exact value of each expression. sin⁻¹ (- 1/2)

Verified step by step guidance

1. Understand that \( \sin^{-1} \) refers to the inverse sine function, also known as arcsin, which gives the angle whose sine is the given value.
2. Recognize that the range of the \( \sin^{-1} \) function is \([-\frac{\pi}{2}, \frac{\pi}{2}]\), meaning it returns angles in this interval.
3. Recall that \( \sin(\theta) = -\frac{1}{2} \) corresponds to angles where the sine value is \(-\frac{1}{2}\).
4. Identify the reference angle for \( \sin(\theta) = \frac{1}{2} \), which is \( \frac{\pi}{6} \) or 30 degrees.
5. Determine the angle in the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\) where \( \sin(\theta) = -\frac{1}{2} \), which is \(-\frac{\pi}{6}\).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as sin⁻¹ (arcsin), are used to find the angle whose sine is a given value. For example, sin⁻¹(x) returns an angle θ such that sin(θ) = x. The range of arcsin is restricted to [-π/2, π/2] to ensure it is a function, meaning it can only return one value for each input.

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is fundamental in trigonometry as it provides a geometric interpretation of the sine, cosine, and tangent functions. The coordinates of points on the unit circle correspond to the cosine and sine values of angles, making it essential for understanding the values of trigonometric functions at specific angles.

Quadrants and Angle Values

In trigonometry, the coordinate plane is divided into four quadrants, each affecting the signs of the sine and cosine values. The sine function is negative in the third and fourth quadrants. For sin⁻¹(-1/2), we need to identify the angle in the specified range that corresponds to this sine value, which occurs in the fourth quadrant, specifically at -π/6 or 330°.