Answer: Choice C. [tex]x = \pm\frac{\pi}{6}[/tex]
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Explanation:
Recall that
[tex]\tan(x) = \frac{\sin(x)}{\cos(x)}\\\\[/tex]
So we can say
[tex]\tan(3x) = \frac{\sin(3x)}{\cos(3x)}\\\\[/tex]
The vertical asymptotes occur when the denominator is zero.
If we were to plug in [tex]x = \frac{\pi}{6}[/tex], then we'd have,
[tex]\tan(3x) = \frac{\sin(3x)}{\cos(3x)}\\\\\tan(3*\frac{\pi}{6}) = \frac{\sin(3*\frac{\pi}{6})}{\cos(3*\frac{\pi}{6})}\\\\\tan(\frac{\pi}{2}) = \frac{\sin(\frac{\pi}{2})}{\cos(\frac{\pi}{2})}\\\\\tan(\frac{\pi}{2}) = \frac{1}{0}\\\\\tan(\frac{\pi}{2}) = \text{und}\text{efined}\\\\[/tex]
This shows that one vertical asymptote is at [tex]x = \frac{\pi}{6}[/tex]
Through similar steps, you should find that another vertical asymptote is at [tex]x = -\frac{\pi}{6}[/tex]
We can condense those two equations into [tex]x = \pm\frac{\pi}{6}[/tex]
