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Home » For a hexagon inscribed in a circle, the side length \( s = \) radius \( = 6 \, \textcm \). - 500apps

For a hexagon inscribed in a circle, the side length \( s = \) radius \( = 6 \, \textcm \). - 500apps

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 Question: Compute $ (\cos 72^\circ + i \sin 72^\circ)^5 $, modeling rotational symmetry in a planetary orbit.

📌 Solution: Using De Moivre’s Theorem, $ (\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta) $. Here, $ n = 5 $, $ \theta = 72^\circ $:

📌 The result is $ \boxed{1} $.

📌 Question: In a city map, vectors $ \overrightarrow{OA} = \begin{bmatrix} 2 \\ 1 \end{bmatrix} $ and $ \overrightarrow{OB} = \begin{bmatrix} -1 \\ 3 \end{bmatrix} $ represent street segments. Find $ \overrightarrow{OC} $ such that $ \overrightarrow{OC} = 2\overrightarrow{OA} - 3\overrightarrow{OB} $.

📌 Solution: Compute $ 2\overrightarrow{OA} = \begin{bmatrix} 4 \\ 2 \end{bmatrix} $ and $ 3\overrightarrow{OB} = \begin{bmatrix} -3 \\ 9 \end{bmatrix} $. Subtract:

📌 \overrightarrow{OC} = \begin{bmatrix} 4 - (-3) \\ 2 - 9 \end{bmatrix} = \begin{bmatrix} 7 \\ -7 \end{bmatrix}.

📌 The vector is $ \boxed{\begin{bmatrix} 7 \\ -7 \end{bmatrix}} $.Question: A volcanologist models the amplitude of seismic waves using the function $ f(x) = \sin^2 x + 2\cos x $. Find the range of $ f(x) $ as $ x $ ranges over all real numbers.

📌 Solution: Start by using the identity $ \sin^2 x = 1 - \cos^2 x $ to rewrite $ f(x) $:

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