Find the periods of the following : (i) `|cos x|` (ii) `tan 4x`

Answer

To find the periods of the functions \( | \cos x | \) and \( \tan 4x \), we will analyze each function step by step.

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

1. **Understanding the function**: The function \( \cos x \) has a standard period of \( 2\pi \). This means that the values of \( \cos x \) repeat every \( 2\pi \) radians.

2. **Modulus effect**: The modulus function \( | \cos x | \) affects the graph of \( \cos x \) by reflecting any negative values above the x-axis. This means that all negative values of \( \cos x \) become positive.

3. **Graphing**: When we graph \( | \cos x | \), we notice that the graph will repeat every \( \pi \) radians instead of \( 2\pi \) because the negative parts of the cosine wave are reflected upwards.

4. **Conclusion**: Therefore, the period of \( | \cos x | \) is \( \pi \).

### Step 2: Finding the period of \( \tan 4x \)

1. **Understanding the function**: The function \( \tan x \) has a standard period of \( \pi \). This means that the values of \( \tan x \) repeat every \( \pi \) radians.

2. **Effect of the coefficient**: When we have \( \tan kx \), the period changes based on the coefficient \( k \). The new period is given by the formula:
\[ \text{New Period} = \frac{\pi}{k} \]
In this case, \( k = 4 \).

3. **Calculating the period**: Substituting \( k = 4 \) into the formula gives us:
\[ \text{Period of } \tan 4x = \frac{\pi}{4} \]

4. **Conclusion**: Therefore, the period of \( \tan 4x \) is \( \frac{\pi}{4} \).

### Final Answer:
- The period of \( | \cos x | \) is \( \pi \).
- The period of \( \tan 4x \) is \( \frac{\pi}{4} \).

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