The period of the function `|sin^3(x/2)|+|cos^5(x/5)|` is

Options

A. `2pi`
B. `10pi`
C. `8pi`
D. `5pi`

Explanation

To find the period of the function \( f(x) = |\sin^3(x/2)| + |\cos^5(x/5)| \), we need to determine the periods of each component of the function separately and then find the least common multiple (LCM) of those periods.

### Step-by-Step Solution:

1. **Identify the components of the function:**
The function consists of two parts:
- \( |\sin^3(x/2)| \)
- \( |\cos^5(x/5)| \)

2. **Find the period of \( |\sin^3(x/2)| \):**
- The basic sine function \( \sin(x) \) has a period of \( 2\pi \).
- The function \( \sin(x/2) \) has a period of \( 2\pi \times 2 = 4\pi \) because the coefficient of \( x \) is \( \frac{1}{2} \).
- The function \( |\sin(x/2)| \) retains the same period of \( 4\pi \).
- The function \( \sin^3(x/2) \) also has a period of \( 4\pi \) since raising to a power does not change the period.
- Therefore, the period of \( |\sin^3(x/2)| \) is \( 4\pi \).

3. **Find the period of \( |\cos^5(x/5)| \):**
- The basic cosine function \( \cos(x) \) also has a period of \( 2\pi \).
- The function \( \cos(x/5) \) has a period of \( 2\pi \times 5 = 10\pi \) because the coefficient of \( x \) is \( \frac{1}{5} \).
- The function \( |\cos(x/5)| \) retains the same period of \( 10\pi \).
- The function \( \cos^5(x/5) \) also has a period of \( 10\pi \).
- Therefore, the period of \( |\cos^5(x/5)| \) is \( 10\pi \).

4. **Calculate the overall period of \( f(x) \):**
- Now we have the periods of both components:
- Period of \( |\sin^3(x/2)| = 4\pi \)
- Period of \( |\cos^5(x/5)| = 10\pi \)
- To find the overall period of the function \( f(x) \), we need to find the least common multiple (LCM) of \( 4\pi \) and \( 10\pi \).
- The LCM of \( 4 \) and \( 10 \) is \( 20 \).
- Therefore, the LCM of \( 4\pi \) and \( 10\pi \) is \( 20\pi \).

5. **Conclusion:**
- The period of the function \( f(x) = |\sin^3(x/2)| + |\cos^5(x/5)| \) is \( 20\pi \).

### Final Answer:
The period of the function \( |\sin^3(x/2)| + |\cos^5(x/5)| \) is \( 20\pi \).

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