Given that \( x = 0.5 \), calculate \( \sin^{-1}(x) \) and explain the result.

To calculate \( \sin^{-1}(0.5) \), we need to find the angle \( \theta \) for which \( \sin(\theta) = 0.5 \). The solution to this equation is \( \theta = \frac{\pi}{6} \) radians (or 30 degrees), because this is the angle in the first quadrant where the sine value equals 0.5. This result means that \( \sin^{-1}(0.5) \) returns the principal value, which is always in the range of \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \) in radians. So, \( \sin^{-1}(0.5) = \frac{\pi}{6} \). If you want to see sine in action, picture this: imagine a right triangle where the opposite side to your angle is 1 unit long, and the hypotenuse is 2 units long. The sine of the angle is the ratio of the opposite side to the hypotenuse, which gives us 1/2, reinforcing that our angle \( \frac{\pi}{6} \) (or 30 degrees) is correct. When calculating the inverse sine function, it's crucial to remember that it is not only about finding the right angle, but also about understanding its position on the unit circle. The angle we found, \( \frac{\pi}{6} \), sits beautifully in the first quadrant where both sine and cosine values are positive, confirming our result and completing the circle of understanding!