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Home » n_1 \sin(\theta_1) = n_2 \sin(\theta_2) - Crosslake

n_1 \sin(\theta_1) = n_2 \sin(\theta_2) - Crosslake

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 \(3b + c = 5\),

📌 \(2c - 3a = 7\),

📌 \(-a - 2b = -4\).

📌 From equation 3: \(a = 4 - 2b\). Substitute into equation 2: \(2c - 3(4 - 2b) = 7 \implies 2c - 12 + 6b = 7 \implies 2c + 6b = 19\). From equation 1: \(c = 5 - 3b\). Substitute into \(2c + 6b = 19\): \(2(5 - 3b) + 6b = 10 - 6b + 6b = 10 = 19\), which is a contradiction. Thus, no solution exists.

📌 \boxed{\text{No such vector } \mathbf{v} \text{ exists.}}

📌 Question: Determine the number of real solutions to the equation \(\sin(2x) = \frac{1}{2}$ for \(x \in [0, 2\pi]\).

📌 Solution: The equation \(\sin(2x) = \frac{1}{2}\) has solutions when \(2x = \frac{\pi}{6} + 2\pi n\) or \(2x = \frac{5\pi}{6} + 2\pi n\) for integer \(n\). Solving for \(x\) in \([0, 2\pi]\):

📌 \(x = \frac{\pi}{12} + \pi n\): Solutions are \(\frac{\pi}{12}, \frac{13\pi}{12}\).

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