Step 1: Use the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$. Substitute the given value $\sin \theta = \frac{3}{5}$ into the identity. $$ \left(\frac{3}{5}\right)^2 + \cos^2 \theta = 1 $$ Step 2: Square the term and simplify. $$ \frac{9}{25} + \cos^2 \theta = 1 $$ Step 3: Isolate $\cos^2 \theta$. $$ \cos^2 \theta = 1 - \frac{9}{25} $$ $$ \cos^2 \theta = \frac{25}{25} - \frac{9}{25} $$ $$ \cos^2 \theta = \frac{16}{25} $$ Step 4: Take the square root of both sides to find $\cos \theta$. $$ \cos \theta = \pm\sqrt{\frac{16}{25}} $$ $$ \cos \theta = \pm\frac{4}{5} $$ Step 5: Determine the sign of $\cos \theta$. Since $\theta$ is an acute angle, it lies in the first quadrant ($0^\circ < \theta < 90^\circ$). In the first quadrant, both sine and cosine values are positive. Therefore, $\cos \theta$ must be positive. $$ \cos \theta = \frac{4}{5} $$ The value of $\cos \theta$ is $\boxed{\frac{4}{5}}$.