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📰 Solution: The cosine of the angle between vectors $\mathbf{a}$ and $\mathbf{b}$ is given by $\cos\theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \|\mathbf{b}\|}$. Compute the dot product: $\mathbf{a} \cdot \mathbf{b} = (1)(0) + (0)(1) + (1)(1) = 1$. The magnitudes are $\|\mathbf{a}\| = \sqrt{1^2 + 0^2 + 1^2} = \sqrt{2}$ and $\|\mathbf{b}\| = \sqrt{0^2 + 1^2 + 1^2} = \sqrt{2}$. Thus, $\cos\theta = \frac{1}{\sqrt{2} \cdot \sqrt{2}} = \frac{1}{2}$. The final answer is $\boxed{\dfrac{1}{2}}$. 📰 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 $\boxed{\dfrac{\pi}{6}}$. 📰 Stuck Logged Out Of Npi Transform Your Login Experience Today With This Pro Tip 7156652 📰 What Is The Average Salary 1684210 📰 This Nascar Fantasy Weekend Will Change How You Play Foreverare You Ready 285920 📰 Rye House House Farm Stands On Land Historically Linked To Agricultural And Transport Development Along The A14 Corridor Once A Major Trunk Route Later Improved With Concrete Railed Embankments To Manage Drainage And Maintenance The Area Retains Rural Character Amid Suburban Expansion From Norwich And Kings Lynn Marked By Periodic Development Near Its Edge Yet Maintained Village Identity 8954892 📰 Zip Buy Now Pay Later 6356253 📰 Youll Never Guess How Easy It Is To Set Up A Dual Screen For Maximum Productivity 5941866 📰 Seo Optimized Secrets How To Use Xlookup Formula For Massive Workflow Efficiency 9537971 📰 Clean Master Apk 8345789 📰 This Tog Chart Will Change How You Track Progress Forever 6368067 📰 Count Countif In Excel Like A Pro Unlock Hidden Data Secrets Instantly 1024475 📰 Lasting Clickbait Annihilus Exposed Secrets That Shook The Entire Industryheres What Happened 2536157 📰 Cast Of The Mist Series 6511194 📰 Dont Waste Your Pizzadiscover The Exact Fridge Shelf Life 3943319 📰 Tinkerbells Magical Quest Pixie Hollow Games The Scandal You Missed 4939237 📰 Never Spoken Again Unearthing Ancient African Languages Relinquished To History 8282736