Final answer:
The equation provided incorrectly states tan (\(\pi/3\)) = \(\sqrt{3}/3\), which should be tan (\(\pi/6\)) instead. The solution to tan (x) = \(\sqrt{3}/3\) within the interval [0, 2\(\pi\)] is \(\pi/6\) only.
Step-by-step explanation:
To solve the equation tan (\(\pi/3\)) = \(\sqrt{3}/3\) over the interval of [0, 2\(\pi\)], we first need to understand the basic properties of the tangent function. The tangent of an angle is the ratio of the sine to the cosine of that angle. Moreover, tangent has a period of \(\pi\), meaning its values repeat every \(\pi\) radians.
The equation tan (\(\pi/3\)) is asking for the angles whose tangent is \(\sqrt{3}/3\). We know that tan (\(\pi/3\)) is \(\sqrt{3}\), not \(\sqrt{3}/3\), which is actually tan (\(\pi/6\)). So we are solving for tan (x) = tan (\(\pi/6\)).
Within the interval [0, 2\(\pi\)], the tangent function will have the same value at \(\pi/6\) and its counterpart in the third quadrant, which is \(\pi/6 + \pi = 7\(\pi/6\)). However, since the question specifies tan (\(\pi/3\)) = \(\sqrt{3}/3\), we are looking for the value \(\pi/6\) only as it is the only value in the given interval where tangent gives us \(\sqrt{3}/3\).
Therefore, the correct answer is \(\pi/6\) only, making option a correct.