Solve the equation on the interval $0 \leq \theta< 2 \pi$.
\[
\cos (3 \theta)=1
\]

Step 1 ：The given equation is \(\cos(3 \theta)=1\).

Step 2 ：This equation is in the form of \(\cos(x) = a\), where \(x = 3\theta\) and \(a = 1\).

Step 3 ：The solutions to this equation are the values of \(x\) for which the cosine function equals 1.

Step 4 ：We know that \(\cos(x) = 1\) for \(x = 2n\pi\), where \(n\) is an integer.

Step 5 ：Therefore, we can set \(3\theta = 2n\pi\) and solve for \(\theta\).

Step 6 ：However, we need to consider the interval \(0 \leq \theta<2 \pi\).

Step 7 ：Therefore, we need to find the values of \(n\) that make \(\theta\) fall within this interval.

Step 8 ：The solutions to the equation are \(\theta = 0\) and \(\theta = 2.09439510239320\).

Step 9 ：These are the values of \(\theta\) that satisfy the equation \(\cos(3\theta) = 1\) within the interval \(0 \leq \theta<2 \pi\).

Step 10 ：Final Answer: The solutions to the equation \(\cos(3\theta) = 1\) on the interval \(0 \leq \theta<2 \pi\) are \(\boxed{0}\) and \(\boxed{2.09439510239320}\).