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Home » Last Stand Review: It Ends With Us Now Forever? Don’t Miss It! - Tacotoon

Last Stand Review: It Ends With Us Now Forever? Don’t Miss It! - Tacotoon

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 Question: In spherical coordinates, describe the surface defined by $\rho = 2\sin\phi \cos\theta$.

📌 Solution: Convert to Cartesian coordinates using $\rho = \sqrt{x^2 + y^2 + z^2}$, $x = \rho \sin\phi \cos\theta$, $y = \rho \sin\phi \sin\theta$, $z = \rho \cos\phi$. Multiply both sides by $\rho$:

📌 \rho^2 = 2\rho \sin\phi \cos\theta \implies x^2 + y^2 + z^2 = 2x.

📌 Rearranging:

📌 x^2 - 2x + y^2 + z^2 = 0 \implies (x^2 - 2x + 1) + y^2 + z^2 = 1 \implies (x - 1)^2 + y^2 + z^2 = 1.

📌 This is a sphere centered at $(1, 0, 0)$ with radius 1.

📌 \boxed{\text{A sphere of radius 1 centered at } (1, 0, 0)}

📌 Question: Find the maximum value of $\left(\sin x + \csc x\right)^2 + \left(\cos x + \sec x\right)^2$ for $0 < x < \frac{\pi}{2}$.

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