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 \(\boxed{\left\{\frac{\pi}{12}, \frac{5\pi}{12}, \frac{3\pi}{4}, \frac{13\pi}{12}, \frac{17\pi}{12}, \frac{7\pi}{4}\right\}}\).