5.4 Determine the general solution of: \( \cos 2 x-\frac{1}{3}=\frac{1}{3} \sin x \)

Solve the equation by following steps: - step0: Solve for \(x\): \(\cos\left(2x\right)-\frac{1}{3}=\frac{1}{3}\sin\left(x\right)\) - step1: Rewrite the expression: \(2\cos^{2}\left(x\right)-1-\frac{1}{3}=\frac{1}{3}\sin\left(x\right)\) - step2: Expand the expression: \(2\cos^{2}\left(x\right)-\frac{4}{3}=\frac{1}{3}\sin\left(x\right)\) - step3: Rewrite the expression: \(\frac{2}{3}-2\sin^{2}\left(x\right)=\frac{1}{3}\sin\left(x\right)\) - step4: Move the expression to the left side: \(\frac{2}{3}-2\sin^{2}\left(x\right)-\frac{1}{3}\sin\left(x\right)=0\) - step5: Factor the expression: \(\frac{1}{3}\left(1-2\sin\left(x\right)\right)\left(2+3\sin\left(x\right)\right)=0\) - step6: Elimination the left coefficient: \(\left(1-2\sin\left(x\right)\right)\left(2+3\sin\left(x\right)\right)=0\) - step7: Separate into possible cases: \(\begin{align}&1-2\sin\left(x\right)=0\\&2+3\sin\left(x\right)=0\end{align}\) - step8: Solve the equation: \(\begin{align}&x=\left\{ \begin{array}{l}\frac{\pi }{6}+2k\pi \\\frac{5\pi }{6}+2k\pi \end{array}\right.,k \in \mathbb{Z}\\&x=\left\{ \begin{array}{l}-\arcsin\left(\frac{2}{3}\right)+2k\pi \\\arcsin\left(\frac{2}{3}\right)+\pi +2k\pi \end{array}\right.,k \in \mathbb{Z}\end{align}\) - step9: Find the union: \(x=\left\{ \begin{array}{l}-\arcsin\left(\frac{2}{3}\right)+2k\pi \\\frac{\pi }{6}+2k\pi \\\frac{5\pi }{6}+2k\pi \\\arcsin\left(\frac{2}{3}\right)+\pi +2k\pi \end{array}\right.,k \in \mathbb{Z}\) To determine the general solution of the equation \[ \cos(2x) - \frac{1}{3} = \frac{1}{3} \sin(x), \] we can rearrange it to: \[ \cos(2x) = \frac{1}{3} \sin(x) + \frac{1}{3}. \] Now, we can solve for \(x\). The solutions are given by: \[ x = \left\{ \begin{array}{l} -\arcsin\left(\frac{2}{3}\right) + 2k\pi \\ \frac{\pi}{6} + 2k\pi \\ \frac{5\pi}{6} + 2k\pi \\ \arcsin\left(\frac{2}{3}\right) + \pi + 2k\pi \end{array} \right., \quad k \in \mathbb{Z}. \] This means that the general solutions for \(x\) are: 1. \(x = -\arcsin\left(\frac{2}{3}\right) + 2k\pi\) 2. \(x = \frac{\pi}{6} + 2k\pi\) 3. \(x = \frac{5\pi}{6} + 2k\pi\) 4. \(x = \arcsin\left(\frac{2}{3}\right) + \pi + 2k\pi\) where \(k\) is any integer.