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

To calculate \( \sin^{-1}(0.5) \), we'll determine the angle whose sine is 0.5. The inverse sine function, denoted as \( \sin^{-1}(x) \) or \( \arcsin(x) \), returns the angle \( \theta \) within the principal range of \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) radians (or \(-90^\circ\) to \(90^\circ\)) such that: \[ \sin(\theta) = x \] ### Calculation: Given: \[ x = 0.5 \] We need to find \( \theta = \sin^{-1}(0.5) \). Recall that: \[ \sin\left(\frac{\pi}{6}\right) = \sin(30^\circ) = 0.5 \] Therefore: \[ \sin^{-1}(0.5) = \frac{\pi}{6} \text{ radians} \quad \text{or} \quad 30^\circ \] ### Explanation: The sine function relates an angle in a right-angled triangle to the ratio of the length of the opposite side to the hypotenuse. Specifically: \[ \sin(\theta) = \frac{\text{Opposite Side}}{\text{Hypotenuse}} \] When we take the inverse sine of a value, we're essentially asking: "What angle \( \theta \) has a sine equal to this value?" For \( x = 0.5 \), the angle \( \theta = 30^\circ \) (or \( \frac{\pi}{6} \) radians) satisfies this condition because: \[ \sin(30^\circ) = \frac{1}{2} = 0.5 \] Thus, \( \sin^{-1}(0.5) = 30^\circ \) or \( \frac{\pi}{6} \) radians. ### Summary: \[ \sin^{-1}(0.5) = 30^\circ = \frac{\pi}{6} \text{ radians} \] This means that an angle of 30 degrees (or \( \frac{\pi}{6} \) radians) has a sine value of 0.5 within the principal range of the inverse sine function.