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📰 Question: A science policy analyst models the efficiency of a renewable energy grid using complex numbers. If $z = \cos\theta + i\sin\theta$ satisfies $z^6 = -1$, find $\theta$ in radians. 📰 Solution: By De Moivre's Theorem, $z^6 = \cos(6\theta) + i\sin(6\theta) = -1 + 0i$. This implies $6\theta = \pi + 2\pi k$ for integer $k$. Solving for $\theta$ gives $\theta = \frac{\pi}{6} + \frac{\pi k}{3}$. The principal solution in $[0, 2\pi)$ is $\theta = \frac{\pi}{6}, \frac{\pi}{2}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{3\pi}{2}, \frac{11\pi}{6}$. The smallest positive $\theta$ is $\boxed{\dfrac{\pi}{6}}$. 📰 Question: A tech entrepreneur develops a sensor system where two unit vectors $\mathbf{p}$ and $\mathbf{q}$ form a plane. If $\mathbf{p} \times \mathbf{q} = \begin{pmatrix} 0 \\ 0 \\ \frac{1}{2} \end{pmatrix}$, find the angle between $\mathbf{p}$ and $\mathbf{q}$. 📰 Hotel En Chandler 7332078 📰 Can You Survive 24 Hours A Day Heres What Happens To Your Body 5471489 📰 The Ultimate Guide To Mastering Jav Web Projects In 2024 5577113 📰 How Old Is Kathy Hilton 6613604 📰 Crucifix Crucifix The Hidden Symbol Driving Controversy In Modern Faith 5672738 📰 The One Chatapp Everyones Usingsee How It Boosts Productivity Instantly 101341 📰 The Final Hour At Kennedy Nightclub Blood Lies And The Night The City Trembled 1930039 📰 You Wont Believe What All Dbz Movies Got Wrong The Hidden Truth Behind Dragon Ball 5496187 📰 Wells Fargo Update 6272399 📰 Blake Livelys Secret Height Shocked Everyoneyou Wont Believe Why 114498 📰 Cell Membrane Structure 5408063 📰 Bugais 8053787 📰 Surprise What Obj Actually Meansyou Wont Believe The Hidden Truth 1894835 📰 Powerball Oklahoma Thriller How One Desperate Bet Changed Everything Overnight 4134420 📰 Breakthrough Access To Specialty Drugs At Costcowatch Your Bill Plummet 5650256