Solve $5 \cos (2 x)=3$ for the smallest three positive solutions.

Give your answers accurate to at least 3 decimal places, as a list separated by commas

Question Help: $\square$ Video

Step 1 ：The given equation is \(5 \cos (2 x)=3\). We need to solve this equation for the smallest three positive solutions.

Step 2 ：First, we can isolate the cosine function by dividing both sides by 5. This gives us \(\cos (2 x) = \frac{3}{5}\).

Step 3 ：Next, we can use the inverse cosine function, also known as arccos, to solve for \(2x\). This gives us \(2x = \arccos(\frac{3}{5})\).

Step 4 ：However, we need to remember that the cosine function is periodic with a period of \(2\pi\). This means that there are infinitely many solutions to this equation. To find the smallest three positive solutions, we can add multiples of \(2\pi\) to the solution we found.

Step 5 ：Finally, we need to divide our solutions by 2 to solve for \(x\) instead of \(2x\).

Step 6 ：The smallest three positive solutions to the equation \(5 \cos (2 x)=3\) are \(\boxed{0.464}\), \(\boxed{3.605}\), and \(\boxed{6.747}\).