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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 $\mathbf{q}$. 📰 Airdrop Not Working 4574696 📰 Finally The Simple Hack That Makes Ready Sub Unstoppable 754479 📰 Heard Museum 4904761 📰 Hotels In Tallahassee Fl 1421859 📰 Wells Fargo Built 835616 📰 Killer Elite Cast 5064233 📰 5Dont Miss It Epd Stock Symbol Is Shaping The Future Of Tech Investing 6427566 📰 Cheap Car Insurance In Ny Scammers Avoid This Hidden Gem At All Costs 6604671 📰 Cast Of Mission Impossible Final Reckoning 6836694 📰 Mickey Ds Coney Island 8742998 📰 Jw Broadcasting Exposed The Hidden Truth Behind Their Massive Reach 9252082 📰 Campaign Synonym 3392518 📰 Unlock Your Destiny Discover Your 15 March Sun Signs Hidden Power 7382174 📰 Free Calls Across Borders This Simple Trick Lets You Talk Internationally Now 3242906 📰 Watch Her Risedark Magician Girl Controls The Shadows With These Secret Spells 2227518