Period of `sin.(pi)/(4)[x]+sin.(pix)/(2)+tan.(pi)/(3)+sin.(pit)/(4)` is ________________

Answer

To find the period of the function given in the question, we will analyze each term in the expression:

The function is:
\[ f(x) = \sin\left(\frac{\pi}{4} x\right) + \sin\left(\frac{\pi}{2} x\right) + \tan\left(\frac{\pi}{3}\right) + \sin\left(\frac{\pi}{4} t\right) \]

### Step 1: Identify the periods of each trigonometric function

1. **Period of \( \sin\left(\frac{\pi}{4} x\right) \)**:
- The general formula for the period of \( \sin(kx) \) is \( \frac{2\pi}{k} \).
- Here, \( k = \frac{\pi}{4} \).
- Therefore, the period is:
\[ T_1 = \frac{2\pi}{\frac{\pi}{4}} = 2\pi \cdot \frac{4}{\pi} = 8 \]

2. **Period of \( \sin\left(\frac{\pi}{2} x\right) \)**:
- Here, \( k = \frac{\pi}{2} \).
- Therefore, the period is:
\[ T_2 = \frac{2\pi}{\frac{\pi}{2}} = 2\pi \cdot \frac{2}{\pi} = 4 \]

3. **Period of \( \tan\left(\frac{\pi}{3}\right) \)**:
- The tangent function has a period of \( \pi \), regardless of the coefficient of \( x \) since \( \frac{\pi}{3} \) is a constant.
- Therefore, the period is:
\[ T_3 = \pi \]

4. **Period of \( \sin\left(\frac{\pi}{4} t\right) \)**:
- This term is similar to the first term, so:
\[ T_4 = 8 \]

### Step 2: Find the least common multiple (LCM) of the periods

Now we need to find the least common multiple of the periods \( T_1 = 8 \), \( T_2 = 4 \), and \( T_3 = \pi \).

- The periods in numerical form are \( 8, 4, \) and \( \pi \).
- The LCM of \( 8 \) and \( 4 \) is \( 8 \).
- To include \( \pi \), we need to express \( 8 \) in terms of a common denominator with \( \pi \).

### Step 3: Calculate LCM

To find the LCM of \( 8 \) and \( \pi \):
- The LCM of \( 8 \) and \( \pi \) is \( 8\pi \) since \( 8 \) is not a multiple of \( \pi \).

### Final Step: Conclusion

Thus, the overall period of the function \( f(x) \) is:
\[ \text{Period} = 8\pi \]

### Answer:
The period of the function is \( 8\pi \).

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