Find the exact value of each expression. sin⁻¹ 1/2

Verified step by step guidance

1. Recognize that the expression \( \sin^{-1} \frac{1}{2} \) represents the inverse sine (arcsine) function, which gives the angle whose sine is \( \frac{1}{2} \).
2. Recall the range of the inverse sine function, which is \( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \), meaning the angle you find must lie within this interval.
3. Identify the angle \( \theta \) such that \( \sin \theta = \frac{1}{2} \) within the principal range. Use your knowledge of special angles on the unit circle.
4. Remember that \( \sin \frac{\pi}{6} = \frac{1}{2} \), so \( \theta = \frac{\pi}{6} \) is the angle that satisfies the equation within the allowed range.
5. Conclude that \( \sin^{-1} \frac{1}{2} = \frac{\pi}{6} \) as the exact value of the expression.

Key Concepts

Inverse Sine Function (sin⁻¹ or arcsin)

The inverse sine function, denoted as sin⁻¹ or arcsin, returns the angle whose sine value is a given number. It is defined for inputs between -1 and 1 and outputs angles in the range [-π/2, π/2] or [-90°, 90°]. Understanding this helps find the angle corresponding to a sine value.

Exact Values of Sine for Special Angles

Certain angles have well-known sine values, such as sin(30°) = 1/2 or sin(π/6) = 1/2. Recognizing these special angles allows you to determine the exact angle for a given sine value without a calculator.

Domain and Range Restrictions of Inverse Trigonometric Functions

Inverse trig functions have restricted domains and ranges to ensure they are functions. For sin⁻¹, the input must be between -1 and 1, and the output angle lies between -90° and 90°. This restriction ensures a unique angle is returned for each sine value.