what is the period of f(x) = sin(x)?\n○ $\frac{pi}{2}$\n○ $pi$\n○ $\frac{3pi}{2}$\n○ $2pi$

Explanation:

Step1: Recall period - definition

The period of a function (y = f(x)) is the smallest positive number (T) such that (f(x + T)=f(x)) for all (x) in the domain of (f). For the sine - function (y = \sin(x)), we know that (\sin(x + 2\pi)=\sin(x)) for all real - valued (x). We also check if there is a smaller positive number. Let's assume there exists a positive number (T<2\pi) such that (\sin(x + T)=\sin(x)) for all (x). But from the unit - circle definition of the sine function, the sine of an angle corresponds to the (y) - coordinate of a point on the unit circle. The pattern of the (y) - coordinates of points on the unit circle repeats every (2\pi) radians.

Step2: Confirm the period

Since there is no positive number (T<2\pi) for which (\sin(x + T)=\sin(x)) for all (x), the period of (y = \sin(x)) is (2\pi).

Answer:

D. (2\pi)