\(x = \frac{\pi}{12} + \pi n\): Solutions are \(\frac{\pi}{12}, \frac{13\pi}{12}\).

February 22, 2026

$\(x = \frac{\pi}{12} + \pi n\): Solutions are \(\frac{\pi}{12}, \frac{13\pi}{12}\).$

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\boxed{\text{No such vector } \mathbf{v} \text{ exists.}}
Question: Determine the number of real solutions to the equation \(\sin(2x) = \frac{1}{2}$ for \(x \in [0, 2\pi]\).
Solution: The equation \(\sin(2x) = \frac{1}{2}\) has solutions when \(2x = \frac{\pi}{6} + 2\pi n\) or \(2x = \frac{5\pi}{6} + 2\pi n\) for integer \(n\). Solving for \(x\) in \([0, 2\pi]\):
\(x = \frac{5\pi}{12} + \pi n\): Solutions are \(\frac{5\pi}{12}, \frac{17\pi}{12}\).
Total of 4 solutions.
Question: If \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\) are unit vectors, find the maximum value of \(\mathbf{u} \cdot (\mathbf{v} + \mathbf{w})\).

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