Problem

Solve the equation for exact solutions over the interval $[0,2 \pi)$.
\[
8 \tan 3 x=8
\]
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solutionset is
(Type an exact answer, using $\pi$ as needed. Type your answer in radians. Use integers or
B. The solution is the empty set.

Answer

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Answer

Final Answer: The solution set is \(\boxed{\left\{\frac{\pi}{12}, \frac{5\pi}{12}, \frac{3\pi}{4}, \frac{13\pi}{12}, \frac{17\pi}{12}, \frac{7\pi}{4}\right\}}\).

Steps

Step 1 :The given equation is \(8 \tan 3 x=8\).

Step 2 :We can simplify this equation by dividing both sides by 8 to get \(\tan 3x = 1\).

Step 3 :The solutions to this equation are the values of \(x\) for which \(3x\) is an odd multiple of \(\frac{\pi}{4}\) in the interval \([0, 2\pi)\).

Step 4 :We can find these values by dividing the odd multiples of \(\frac{\pi}{4}\) by 3.

Step 5 :The solutions to the equation are \(x = \frac{\pi}{12}, \frac{5\pi}{12}, \frac{3\pi}{4}, \frac{13\pi}{12}, \frac{17\pi}{12}, \frac{7\pi}{4}\).

Step 6 :These are the values of \(x\) for which \(\tan 3x = 1\) in the interval \([0, 2\pi)\).

Step 7 :Final Answer: The solution set is \(\boxed{\left\{\frac{\pi}{12}, \frac{5\pi}{12}, \frac{3\pi}{4}, \frac{13\pi}{12}, \frac{17\pi}{12}, \frac{7\pi}{4}\right\}}\).

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