Question

The period of the function \( f(x) = \sin\left( \frac{3x}{2} \right) \) is equal to:

• A. \( \frac{4\pi}{3} \)
• B. \( \frac{2\pi}{3} \)
• C. \( \frac{\pi}{3} \)
• D. \( 3\pi \)
• E. \( 2\pi \)

Hint:

For sinusoidal functions of the form \( f(x) = \sin(kx) \), the period is always \( \frac{2\pi}{|k|} \). This formula can be applied directly to determine the period of any sine function.

Answer 1

The Correct Option is A

The general form for the period of a sine function \( f(x) = \sin(kx) \) is given by: \[ {Period} = \frac{2\pi}{|k|} \] For the function \( f(x) = \sin\left( \frac{3x}{2} \right) \), the coefficient \( k = \frac{3}{2} \).
Substituting \( k \) into the period formula: \[ {Period} = \frac{2\pi}{\left|\frac{3}{2}\right|} = \frac{2\pi}{\frac{3}{2}} = \frac{4\pi}{3} \]
Thus, the period of the function is \( \frac{4\pi}{3} \).