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Home » Drastic Improvement? How Duol & Yahoo Finance Transform Your Daily Learning! - Coaching Toolbox

Drastic Improvement? How Duol & Yahoo Finance Transform Your Daily Learning! - Coaching Toolbox

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Solution: The equation $ \sin(2\theta) = \frac{\sqrt{3}}{2} $ implies $ 2\theta = 60^\circ + 360^\circ k $ or $ 2\theta = 120^\circ + 360^\circ k $ for integer $ k $. Solving for $ \theta $, we get $ \theta = 30^\circ + 180^\circ k $ or $ \theta = 60^\circ + 180^\circ k $. Within $ [0^\circ, 360^\circ] $, the solutions are $ 30^\circ, 60^\circ, 210^\circ, 240^\circ $. Thus, the answer is $ \boxed{30^\circ}, \boxed{60^\circ}, \boxed{210^\circ}, \boxed{240^\circ} $.

📌 Question: Determine the matrix $ \mathbf{M} $ such that $ \mathbf{M} \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} $.

📌 Solution: Let $ \mathbf{M} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} $. Multiplying $ \mathbf{M} $ with $ \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} $ gives:

📌 \begin{pmatrix} a(1) + b(3) & a(2) + b(4) \\ c(1) + d(3) & c(2) + d(4) \end{pmatrix} = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}.

📌 This leads to the system:

📌 $ a + 3b = 5 $,

📌 $ 2a + 4b = 6 $,

📌 $ c + 3d = 7 $,

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