Find the vertical asymptotes (if any) of the graph of the function. (Use \( n \) as an arbitrary integer if necessary. If an answer does not exist, enter DNE.) \[ g(\theta)=\frac{\tan (6 \theta)}{5 \theta} \] \( \pi \quad n \pi \)

To find the vertical asymptotes of the function \( g(\theta) = \frac{\tan(6\theta)}{5\theta} \), we need to identify where the function becomes undefined. This can happen when the denominator equals zero or when the tangent function approaches infinity. 1. **Denominator Analysis**: The denominator \( 5\theta \) is zero when \( \theta = 0 \). This creates a vertical asymptote since the function approaches infinity (or negative infinity) as \(\theta\) approaches 0. 2. **Tangent Function Analysis**: The tangent function \( \tan(6\theta) \) has vertical asymptotes where \( 6\theta = \frac{\pi}{2} + n\pi \) for any integer \( n \). Solving for \(\theta\) gives: \[ \theta = \frac{\pi/2 + n\pi}{6} = \frac{\pi}{12} + \frac{n\pi}{6} \] This shows additional vertical asymptotes occurring at \( \theta = \frac{\pi}{12} + \frac{n\pi}{6} \). Thus, the vertical asymptotes of the function are: \[ \theta = 0, \quad \frac{\pi}{12} + \frac{n\pi}{6} \quad \text{for any integer } n. \] This can be noted more compactly as \( \theta = 0 \) and \( \theta = \frac{\pi}{12} + \frac{n\pi}{6} \), capturing all instances where the function becomes undefined.