Find the vertical asymptotes (if any) of the graph of the function. (Üe a as an arbitrary Integer if necessary. If an answer does not exist, enter DNE.) \[ \begin{array}{l} g(\theta)=\frac{\tan (6 \theta)}{5 \theta} \\ \theta=\square \end{array} \] Need Help? Aload lit Master 4

To find the vertical asymptotes of the function \( g(\theta) = \frac{\tan(6\theta)}{5\theta} \), we need to determine where the function is undefined. The vertical asymptotes occur where the denominator is zero or where the tangent function is undefined. First, the denominator \( 5\theta \) is zero when \( \theta = 0 \). This gives us one vertical asymptote at \( \theta = 0 \). Next, the tangent function \( \tan(6\theta) \) is undefined for values where \( 6\theta = \frac{\pi}{2} + n\pi \) for any integer \( n \). Solving for \( \theta \), we get: \[ \theta = \frac{\pi}{12} + \frac{n\pi}{6} \] These values, including \( \theta = 0 \), yield the vertical asymptotes. So, the complete set of vertical asymptotes can be expressed as: \[ \theta = 0, \quad \theta = \frac{\pi}{12} + \frac{n\pi}{6} \] for integers \( n \). In brief, the vertical asymptotes are located at the specified values of \( \theta \) where the function is undefined.