solve the following equation on the interval 0,2π).\n\\( \\tan ( 3 x ) = \\frac { \\sqrt { 3 } } { 3 }…

solve the following equation on the interval 0,2π).\n\\( \\tan ( 3 x ) = \\frac { \\sqrt { 3 } } { 3 } \\)\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. \\( x = \\) (type an exact answer, using \\( \\pi \\) as needed. use a comma to separate answers as needed. type your answer in\nradians. simplify your answer. use integers or fractions for any numbers in the expression.)\nb. there is no solution.

solve the following equation on the interval 0,2π).\n\\( \\tan ( 3 x ) = \\frac { \\sqrt { 3 } } { 3 } \\)\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. \\( x = \\) (type an exact answer, using \\( \\pi \\) as needed. use a comma to separate answers as needed. type your answer in\nradians. simplify your answer. use integers or fractions for any numbers in the expression.)\nb. there is no solution.

Answer

Explanation:

Step1: Recall the general solution of (\tan\theta = k)

The general solution of (\tan\theta=\frac{\sqrt{3}}{3}) is (\theta = n\pi+\frac{\pi}{6},n\in\mathbb{Z}). Here (\theta = 3x), so (3x=n\pi+\frac{\pi}{6}).

Step2: Solve for (x)

Divide both sides of the equation (3x=n\pi+\frac{\pi}{6}) by (3). We get (x=\frac{n\pi}{3}+\frac{\pi}{18}).

Step3: Find solutions in the interval ([0,2\pi))

When (n = 0): (x=\frac{\pi}{18}) When (n = 1): (x=\frac{\pi}{3}+\frac{\pi}{18}=\frac{6\pi + \pi}{18}=\frac{7\pi}{18}) When (n = 2): (x=\frac{2\pi}{3}+\frac{\pi}{18}=\frac{12\pi+\pi}{18}=\frac{13\pi}{18}) When (n = 3): (x=\pi+\frac{\pi}{18}=\frac{19\pi}{18}) When (n = 4): (x=\frac{4\pi}{3}+\frac{\pi}{18}=\frac{24\pi+\pi}{18}=\frac{25\pi}{18}) When (n = 5): (x=\frac{5\pi}{3}+\frac{\pi}{18}=\frac{30\pi+\pi}{18}=\frac{31\pi}{18})

Answer:

(x = \frac{\pi}{18},\frac{7\pi}{18},\frac{13\pi}{18},\frac{19\pi}{18},\frac{25\pi}{18},\frac{31\pi}{18})