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📰 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 $\mathbf{q}$. 📰 Solution: The magnitude of the cross product is $\|\mathbf{p} \times \mathbf{q}\| = \|\mathbf{p}\| \|\mathbf{q}\| \sin\theta$. Since $\mathbf{p}$ and $\mathbf{q}$ are unit vectors, $\|\mathbf{p} \times \mathbf{q}\| = \sin\theta = \frac{1}{2}$. Thus, $\theta = \arcsin\left(\frac{1}{2}\right)$, which gives $\theta = \frac{\pi}{6}$ or $\frac{5\pi}{6}$. However, the angle between vectors is taken as the acute angle, so $\theta = \boxed{\dfrac{\pi}{6}}$. 📰 What The Check Engine Light Flashing Really Meansdont Ignore This Urgent Alert 3857501 📰 Relive The Horror Unbelievable Scary Sounds That Send Shivers Down Spines 7942019 📰 Ambivalent Meaning 6408452 📰 Water Filter Shower Head 4865719 📰 How To Change Pointer Color 📰 Project Gunship 📰 Mailing Overseas Usps 8053391 📰 Homewood Suites Nyc 8556493 📰 Actor Jerry Ferrara 674290 📰 Credit Acceptance Stock 📰 Compute The Sum Of The Real Parts Of All Complex Numbers Z Satisfying Z4 4Z2 4 0 6837783 📰 This Secret Ingredient In Champs Chicken Changed Everything 1712329 📰 Mathsspotcpm 6005317 📰 How To Cancel Verizon Wireless Order 8248587 📰 Major Update Verizon Internet Vs Xfinity And The World Is Watching