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Determine the general solution of: \( \sin 2 x=4 \cos 2 x \)

Ask by Huff Sandoval. in South Africa
Mar 08,2025

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The general solution is \( x = \frac{\arctan(4)}{2} + \frac{k\pi}{2} \), where \( k \) is any integer.

Solución

Solve the equation \( \sin(2x)=4\cos(2x) \). Solve the equation by following steps: - step0: Solve for \(x\): \(\sin\left(2x\right)=4\cos\left(2x\right)\) - step1: Divide both sides: \(\frac{\sin\left(2x\right)}{\cos\left(2x\right)}=4\) - step2: Divide the terms: \(\tan\left(2x\right)=4\) - step3: Use the inverse trigonometric function: \(2x=\arctan\left(4\right)\) - step4: Add the period: \(2x=\arctan\left(4\right)+k\pi ,k \in \mathbb{Z}\) - step5: Solve the equation: \(x=\frac{\arctan\left(4\right)}{2}+\frac{k\pi }{2},k \in \mathbb{Z}\) The general solution of the equation \( \sin 2x = 4 \cos 2x \) is given by \( x = \frac{\arctan(4)}{2} + \frac{k\pi}{2} \), where \( k \) is an integer.

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The Deep Dive

To solve the equation \( \sin 2x = 4 \cos 2x \), we can start by dividing both sides by \( \cos 2x \) (assuming \( \cos 2x \neq 0 \)), leading to: \[ \tan 2x = 4 \] Next, we find the general solution for \( 2x \): \[ 2x = \tan^{-1}(4) + n\pi, \quad n \in \mathbb{Z} \] Now, divide by 2 to solve for \( x \): \[ x = \frac{1}{2} \tan^{-1}(4) + \frac{n\pi}{2}, \quad n \in \mathbb{Z} \] Thus, the general solution is: \[ x = \frac{1}{2} \tan^{-1}(4) + \frac{n\pi}{2}, \quad n \in \mathbb{Z} \]

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