Solve the equation for exact solutions over the linterval \( [0,2 \pi) \). 7 tan \( 3 x=7 \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is \( \{ \). Type an exact answer, using \( \pi \) as needed. Type your answer in radians Use integers or fractio for any numbers in the expression. Use a comma to separate answers as needed.) B. The solution is the empty set.
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To solve the equation \( 7 \tan(3x) = 7 \), we can first divide both sides by 7 (provided \( 7 \neq 0 \)), yielding \( \tan(3x) = 1 \). The general solutions for \( \tan(\theta) = 1 \) are \( \theta = \frac{\pi}{4} + n\pi \), where \( n \) is any integer. Now, substituting \( \theta = 3x \) gives us \( 3x = \frac{\pi}{4} + n\pi \). Dividing by 3 gives the solutions for \( x \): \[ x = \frac{\pi}{12} + \frac{n\pi}{3} \] Next, we need to find the specific values of \( x \) that lie within the interval \( [0, 2\pi) \): 1. For \( n = 0 \): \[ x = \frac{\pi}{12} \] 2. For \( n = 1 \): \[ x = \frac{\pi}{12} + \frac{\pi}{3} = \frac{\pi}{12} + \frac{4\pi}{12} = \frac{5\pi}{12} \] 3. For \( n = 2 \): \[ x = \frac{\pi}{12} + \frac{2\pi}{3} = \frac{\pi}{12} + \frac{8\pi}{12} = \frac{9\pi}{12} = \frac{3\pi}{4} \] 4. For \( n = 3 \): \[ x = \frac{\pi}{12} + \pi = \frac{\pi}{12} + \frac{12\pi}{12} = \frac{13\pi}{12} \] 5. For \( n = 4 \): \[ x = \frac{\pi}{12} + \frac{5\pi}{3} = \frac{\pi}{12} + \frac{20\pi}{12} = \frac{21\pi}{12} \] (which is greater than \( 2\pi \)) So the valid solutions are: \[ x = \frac{\pi}{12}, \frac{5\pi}{12}, \frac{3\pi}{4}, \frac{13\pi}{12} \] Thus, the solution set is \( \left\{ \frac{\pi}{12}, \frac{5\pi}{12}, \frac{3\pi}{4}, \frac{13\pi}{12} \right\} \).
