The equation given is cos 3x = 1/2. To find the general solution, we need to determine the values of x that satisfy this equation. The general solution will provide a range of values for x that make the equation true.

Step 1: Finding the Reference Angle
To solve the equation, we start by finding the reference angle. The reference angle is the acute angle formed between the terminal side of an angle and the x-axis.

Step 2: Applying the Cosine Function
Since the cosine function repeats itself every 2π radians (or 360 degrees), we can establish the general solution by finding the values of x that satisfy the equation within this period. We can use the reference angle to determine the values of x.

Step 3: Solving for x
To solve the equation cos 3x = 1/2, we need to find the values of x that satisfy this equation. Here's how we can do it:

1. Find the reference angle θ by taking the inverse cosine of 1/2:
θ = cos^(-1)(1/2) = π/3

2. Determine the values of x within a 2π (or 360 degrees) period that satisfy the equation:
- First solution: 3x = π/3
x = π/9

- Second solution: 3x = 2π - π/3
x = (2π - π/3)/3 = (6π - π)/9 = 5π/9

- Third solution: 3x = 2π + π/3
x = (2π + π/3)/3 = (6π + π)/9 = 7π/9

3. Continue finding solutions by adding or subtracting multiples of 2π/3:
- Adding 2π/3 to the previous solution:
x = π/9 + 2π/3 = 7π/9

- Adding 2π/3 to the previous solution:
x = 5π/9 + 2π/3 = 9π/9 = π

- Adding 2π/3 to the previous solution:
x = 7π/9 + 2π/3 = 11π/9

General Solution:
The general solution for cos 3x = 1/2 is:
x = π/9 + (2π/3)n, where n is an integer.

This means that any value of x that can be expressed as π/9 plus a multiple of 2π/3 will satisfy the equation cos 3x = 1/2.

Summary:
- The reference angle is found by taking the inverse cosine of 1/2, which equals π/3.
- By solving the equation within a 2π (or 360 degrees) period and adding or subtracting multiples of 2π/3, we find the general solution for x.
- The general solution is x = π/9 + (2π/3)n, where n is an integer.