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Understanding trigonometric values is foundational for solving problems in trigonometry. These values represent the ratios of the sides of a right triangle when compared to one of the non-right angles within the triangle. The primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan), and they are associated with specific angle measures.

For instance, certain angles have well-known trigonometric values that can be memorized for quick recall. These angles are commonly 30°, 45°, 60°, and their radian equivalents, which are fractions of \( \pi \). The sine of 30° or \( \frac{\pi}{6} \) radians, for example, is \( \frac{1}{2} \) as shown in the exercise. This information is critical for analyzing and understanding functions and their inverses, as well as for solving various trigonometric equations without the need for a calculator.

Trigonometry frequently involves dealing with a set of common angles, which are typically measured in degrees or radians. Angles such as 0°, 30°, 45°, 60°, 90° and their respective radian measures \( 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2} \) are considered 'special' because their trigonometric values are simple and can be easily remembered. These specific angles form the foundation of many trigonometric properties and theorems.

Understanding these angles helps in simplifying the process of evaluating trigonometric expressions, as recognizing the value of trigonometric functions at these common angles can lead straight to the answer. When confronted with a problem requiring the evaluation of a trigonometric expression, recalling these common angle values is a very effective technique.

The arcsine function, often denoted as \( \arcsin \), is the inverse of the sine function. It takes a number between -1 and 1, which represents a sine value, and returns the corresponding angle measure where the sine of that angle is equal to the number provided. For the arcsine of \( \frac{1}{2} \), the answer is the angle whose sine is \( \frac{1}{2} \). In our case, as we have already identified from common angles, the sine of 30° or \( \frac{\pi}{6} \) radians is \( \frac{1}{2} \) which makes \( \arcsin(\frac{1}{2}) \) equals to 30° or \( \frac{\pi}{6} \) radians.

It is essential to remember that the output of the arcsine function is an angle -- and in the context of a right-angled triangle, it corresponds to the angle opposite the side of the length that was used as the input for the function. This enables us to map any trigonometric value back to an angle, which is particularly useful in geometry and solving triangle-based problems.