The period of `sin(pi/4)x+cos(pi/2)x+cos(pi/3)x` is

Explanation

To find the period of the function \( \sin\left(\frac{\pi}{4} x\right) + \cos\left(\frac{\pi}{2} x\right) + \cos\left(\frac{\pi}{3} x\right) \), we will follow these steps:

### Step 1: Identify the periods of each term
The period of the sine and cosine functions can be calculated using the formula:
\[ \text{Period} = \frac{2\pi}{a} \]
where \( a \) is the coefficient of \( x \).

1. For \( \sin\left(\frac{\pi}{4} x\right) \):
- Coefficient \( a = \frac{\pi}{4} \)
- Period = \( \frac{2\pi}{\frac{\pi}{4}} = 2\pi \cdot \frac{4}{\pi} = 8 \)

2. For \( \cos\left(\frac{\pi}{2} x\right) \):
- Coefficient \( a = \frac{\pi}{2} \)
- Period = \( \frac{2\pi}{\frac{\pi}{2}} = 2\pi \cdot \frac{2}{\pi} = 4 \)

3. For \( \cos\left(\frac{\pi}{3} x\right) \):
- Coefficient \( a = \frac{\pi}{3} \)
- Period = \( \frac{2\pi}{\frac{\pi}{3}} = 2\pi \cdot \frac{3}{\pi} = 6 \)

### Step 2: Find the Least Common Multiple (LCM) of the periods
Now, we need to find the LCM of the periods calculated:
- Periods: 8, 4, and 6.

To find the LCM:
- The prime factorization of each number:
- \( 8 = 2^3 \)
- \( 4 = 2^2 \)
- \( 6 = 2^1 \cdot 3^1 \)

The LCM is obtained by taking the highest power of each prime:
- For \( 2 \): \( 2^3 \)
- For \( 3 \): \( 3^1 \)

Thus, the LCM is:
\[ \text{LCM} = 2^3 \cdot 3^1 = 8 \cdot 3 = 24 \]

### Step 3: Conclusion
The period of the function \( \sin\left(\frac{\pi}{4} x\right) + \cos\left(\frac{\pi}{2} x\right) + \cos\left(\frac{\pi}{3} x\right) \) is \( 24 \).

### Final Answer:
The period is \( 24 \).

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