The fundamental trigonometric identity states that \( \sin^2 \theta + \cos^2 \theta = 1 \). Given that \( \sin \theta = \frac{3}{5} \), we substitute this value into the identity: \( \left( \frac{3}{5} \right)^2 + \cos^2 \theta = 1 \), which simplifies to \( \frac{9}{25} + \cos^2 \theta = 1 \). Solving for \( \cos^2 \theta \), we get \( \cos^2 \theta = 1 - \frac{9}{25} \), resulting in \( \cos^2 \theta = \frac{16}{25} \). Therefore, \( \cos \theta = \frac{4}{5} \). The value of \( \cos \theta \) is \( \frac{4}{5} \), corresponding to option (1).