Question

The period of the function \( \sin\left( \frac{\pi x}{4} \right) \) is

• A. 4
• B. \( 4\pi \)
• C. \( 8\pi \)
• D. 8
• E. \( 2\pi \)

Hint:

The period of the sine function \( \sin(kx) \) is \( \frac{2\pi}{|k|} \).

Answer 1

The Correct Option is D

The general form of a sine function is: \[ y = \sin(kx) \] where the period of the sine function is given by: \[ {Period} = \frac{2\pi}{|k|} \] Here, \( k \) is the coefficient of \( x \) in the argument of the sine function. In our case, the function is \( \sin\left( \frac{\pi x}{4} \right) \), so \( k = \frac{\pi}{4} \).
Using the formula for the period, we get: \[ {Period} = \frac{2\pi}{\left|\frac{\pi}{4}\right|} = \frac{2\pi}{\frac{\pi}{4}} = 8 \] Thus, the period of the function \( \sin\left( \frac{\pi x}{4} \right) \) is \( 8 \). Thus, the correct answer is \( \boxed{8} \), corresponding to option (D).