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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 $\mathbf{q}$. 📰 Is This The Most Shocking Final Challenge Of John Cenas Legacy 1119809 📰 What Is A Cash Advance On A Credit Card 6765764 📰 Park City Southie 5238759 📰 Capillary Refill 4752905 📰 Edd Gould 5242311 📰 Gw2 Suarions In Au 9639995 📰 Can You Unlink Fortnite Account From Xbox 8926877 📰 Verizon Internet Gateway Login 3274956 📰 Finally The Best Way To Control Your Canon Camera From Ipad Pro Tips Inside 262579 📰 Actually A Better Approach Use The Formula For Inradius In Terms Of Circumradius But No 4107237 📰 Wells Fargo Business Account Promotion 8882887 📰 How To Cross Out In Excel 8650593 📰 Duane Lee Bounty Hunter 5390208 📰 How To Post On Instagram Anonymously Without Getting Caughtshocking Tips Inside 3297443 📰 Bank Of America Wells Me 5046090