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Home » \left(3, \frac78 \right) - Sterling Industries

\left(3, \frac78 \right) - Sterling Industries

📅 April 16, 2026 👤 scraface

Apr 16, 2026

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📌 This is the only possible value under the constraint, so the maximum is:

📌 \boxed{-\frac{1}{2}}

📌 Question:** A digital transformation algorithm rotates a vector in 3D space using $\tan(\theta)$, where $\theta$ is the angle between two planes. If $\cos \theta = \frac{1}{3}$, compute $\tan \theta$.

📌 Solution:** Use identity:

📌 \tan \theta = \frac{\sqrt{1 - \cos^2 \theta}}{\cos \theta}

📌 Given $\cos \theta = \frac{1}{3}$, compute:

📌 \tan \theta = \frac{\sqrt{1 - \left(\frac{1}{3}\right)^2}}{\frac{1}{3}} = \frac{\sqrt{1 - \frac{1}{9}}}{\frac{1}{3}} = \frac{\sqrt{\frac{8}{9}}}{\frac{1}{3}} = \frac{\frac{2\sqrt{2}}{3}}{\frac{1}{3}} = 2\sqrt{2}

📌 $$Question: A venture capitalist invests in 5 startups, each with a 25% chance of success independently. What is the probability that exactly 2 of the startups succeed?

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