Find the periods of the following functions, if periodic : (i) `| cos x |` (ii) `2 "cos" (1)/(3)(x - pi)`

Answer

To find the periods of the given functions, we will analyze each function step by step.

### Step 1: Finding the period of \( | \cos x | \)

1. **Understanding the function**: The function \( | \cos x | \) is the absolute value of the cosine function. The cosine function itself has a period of \( 2\pi \).

2. **Graphing the function**: The graph of \( \cos x \) oscillates between -1 and 1. When we take the absolute value, all negative values are reflected above the x-axis. This means that the graph will repeat every \( \pi \) instead of \( 2\pi \).

3. **Identifying the period**: Since \( | \cos x | \) has a repeating pattern every \( \pi \), we conclude that the period of \( | \cos x | \) is:
\[ \text{Period} = \pi \]

### Step 2: Finding the period of \( 2 \cos \left( \frac{1}{3}(x - \pi) \right) \)

1. **Identifying the standard form**: The general form of a cosine function is \( k \cos(ax - b) \), where the period is given by \( \frac{2\pi}{|a|} \).

2. **Extracting parameters**: In our function \( 2 \cos \left( \frac{1}{3}(x - \pi) \right) \):
- \( k = 2 \)
- \( a = \frac{1}{3} \)
- \( b = \pi \) (the phase shift does not affect the period)

3. **Calculating the period**: Using the formula for the period:
\[ \text{Period} = \frac{2\pi}{\left| \frac{1}{3} \right|} = 2\pi \times 3 = 6\pi \]

### Final Answers
- The period of \( | \cos x | \) is \( \pi \).
- The period of \( 2 \cos \left( \frac{1}{3}(x - \pi) \right) \) is \( 6\pi \).