The period of the function ` sin""((pix)/(2))+cos((pix)/(2))`, is

Options

A. 4
B. 6
C. 12
D. 24

Answer

To find the period of the function \( f(x) = \sin\left(\frac{\pi x}{2}\right) + \cos\left(\frac{\pi x}{2}\right) \), we need to analyze the individual components of the function.

### Step 1: Identify the periods of the sine and cosine functions

The standard period of the sine function \( \sin(kx) \) is given by:
\[ \text{Period} = \frac{2\pi}{k} \]
Similarly, for the cosine function \( \cos(kx) \), the period is also:
\[ \text{Period} = \frac{2\pi}{k} \]

### Step 2: Calculate the period of \( \sin\left(\frac{\pi x}{2}\right) \)

In our function, \( k = \frac{\pi}{2} \). Therefore, the period of \( \sin\left(\frac{\pi x}{2}\right) \) is:
\[ \text{Period} = \frac{2\pi}{\frac{\pi}{2}} = 2 \cdot 2 = 4 \]

### Step 3: Calculate the period of \( \cos\left(\frac{\pi x}{2}\right) \)

Similarly, for \( \cos\left(\frac{\pi x}{2}\right) \), the period is the same:
\[ \text{Period} = \frac{2\pi}{\frac{\pi}{2}} = 2 \cdot 2 = 4 \]

### Step 4: Determine the overall period of the function

Since both components \( \sin\left(\frac{\pi x}{2}\right) \) and \( \cos\left(\frac{\pi x}{2}\right) \) have the same period of 4, the overall period of the function \( f(x) \) is also 4.

### Conclusion

Thus, the period of the function \( \sin\left(\frac{\pi x}{2}\right) + \cos\left(\frac{\pi x}{2}\right) \) is:
\[ \text{Period} = 4 \]

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