Questions: Find the exact values of s in the interval [0,2 π) that satisfy the given condition sin^2 s = 1/4.

Solution Steps

To solve the equation \(\sin^2(s) = \frac{1}{4}\) for \(s\) in the interval \([0, 2\pi)\), we need to find the values of \(s\) where \(\sin(s) = \pm \frac{1}{2}\). We then identify the angles in the given interval that satisfy this condition.

We start with the equation \(\sin^2(s) = \frac{1}{4}\). This implies that \(\sin(s) = \pm \frac{1}{2}\).

The angles \(s\) in the interval \([0, 2\pi)\) where \(\sin(s) = \frac{1}{2}\) are: \[ s = \frac{\pi}{6}, \quad s = \pi - \frac{\pi}{6} \] Simplifying these, we get: \[ s = \frac{\pi}{6}, \quad s = \frac{5\pi}{6} \]

The angles \(s\) in the interval \([0, 2\pi)\) where \(\sin(s) = -\frac{1}{2}\) are: \[ s = 2\pi - \frac{\pi}{6}, \quad s = \pi + \frac{\pi}{6} \] Simplifying these, we get: \[ s = \frac{11\pi}{6}, \quad s = \frac{7\pi}{6} \]

Final Answer