For which angle measures does the sine and cosine functions exhibit periodicity?

The sine and cosine functions are both periodic functions, meaning they repeat their values at regular intervals. For the sine function, the period is (2\pi), and for the cosine function, the period is also (2\pi). This means that for any angle (x), the following holds true:

• ( \sin(x) = \sin(x + 2k\pi) )

• ( \cos(x) = \cos(x + 2k\pi) )

where (k) is any integer. These equations indicate that sine and cosine will return to the same value whenever you add or subtract (2\pi) multiplied by any integer value.

The options relating to (k) in the other answers involve factors like (\pi) or shifts of (\pi), which do not represent the full periodicity of sine and cosine. For instance, while the sine and cosine functions do exhibit certain symmetries and behaviors at these angles, they do not repeat their cycles fully until you reach a change of (2\pi).

Thus, the correct representation of where sine and cosine functions exhibit periodicity is associated with (2k\pi), indicating that