The period of f(x)=5 sin 3x-7 sin 8x , is

Options

A. `pi`
B. `2pi`
C. `3pi`
D. `8 pi`

Explanation

To find the period of the function \( f(x) = 5 \sin(3x) - 7 \sin(8x) \), we will follow these steps:

### Step 1: Identify the periods of the individual sine functions.
The period of the sine function \( \sin(kx) \) is given by the formula:
\[ \text{Period} = \frac{2\pi}{|k|} \]
where \( k \) is the coefficient of \( x \).

### Step 2: Calculate the period of \( \sin(3x) \).
For \( \sin(3x) \):
- Here, \( k = 3 \).
- Therefore, the period is:
\[ \text{Period of } \sin(3x) = \frac{2\pi}{3} \]

### Step 3: Calculate the period of \( \sin(8x) \).
For \( \sin(8x) \):
- Here, \( k = 8 \).
- Therefore, the period is:
\[ \text{Period of } \sin(8x) = \frac{2\pi}{8} = \frac{\pi}{4} \]

### Step 4: Find the least common multiple (LCM) of the periods.
To find the overall period of \( f(x) \), we need to find the LCM of the two periods calculated:
- The periods are \( \frac{2\pi}{3} \) and \( \frac{\pi}{4} \).

### Step 5: Convert the periods to a common denominator.
The least common multiple of the denominators \( 3 \) and \( 4 \) is \( 12 \). We convert both fractions:
- \( \frac{2\pi}{3} = \frac{8\pi}{12} \)
- \( \frac{\pi}{4} = \frac{3\pi}{12} \)

### Step 6: Calculate the LCM of the numerators.
Now we find the LCM of the numerators \( 8\pi \) and \( 3\pi \):
- The LCM of \( 8 \) and \( 3 \) is \( 24 \), so:
\[ \text{LCM of } \left(\frac{8\pi}{12}, \frac{3\pi}{12}\right) = \frac{24\pi}{12} = 2\pi \]

### Step 7: Find the highest common factor (HCF) of the denominators.
The HCF of the denominators \( 3 \) and \( 4 \) is \( 1 \).

### Step 8: Calculate the overall period.
Thus, the period of \( f(x) \) can be calculated as:
\[ \text{Period of } f(x) = \frac{\text{LCM of the numerators}}{\text{HCF of the denominators}} = \frac{2\pi}{1} = 2\pi \]

### Conclusion
The period of the function \( f(x) = 5 \sin(3x) - 7 \sin(8x) \) is \( 2\pi \).

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