$\cos \theta=\frac{6}{10}, \theta$ lies in quadrant IV

Step 1 ：The cosine function is positive in the fourth quadrant, so the given condition is satisfied. However, we need to find the value of \(\theta\) in degrees. We know that \(\cos \theta = \frac{6}{10}\), so \(\theta = \cos^{-1}(\frac{6}{10})\). But this will give us the angle in the first quadrant. To find the angle in the fourth quadrant, we subtract this angle from \(360^\circ\).

Step 2 ：Calculate \(\theta_1 = \cos^{-1}(\frac{6}{10})\) to get \(\theta_1 = 0.9272952180016123\)

Step 3 ：Convert \(\theta_1\) to degrees to get \(\theta_1_{deg} = 53.13010235415599\)

Step 4 ：Subtract \(\theta_1_{deg}\) from \(360^\circ\) to get the angle in the fourth quadrant, \(\theta_{4_{deg}} = 306.869897645844\)

Step 5 ：Final Answer: \(\theta = \boxed{306.87^\circ}\) in the fourth quadrant.