To determine the maximum value of the function [tex]\( f(x) = \sin(x) \)[/tex], let's analyze the properties of the sine function.
1. Understanding the Sine Function:
- The sine function, sin(x), is a periodic function with a period of [tex]\( 2\pi \)[/tex].
- It oscillates between -1 and 1 for all values of x.
2. Behavior of [tex]\( \sin(x) \)[/tex] Within One Period:
- Over one period, [tex]\( \sin(x) \)[/tex] starts at 0, reaches 1 at [tex]\( \frac{\pi}{2} \)[/tex], comes back to 0 at [tex]\( \pi \)[/tex], reaches -1 at [tex]\( \frac{3\pi}{2} \)[/tex], and completes the cycle at [tex]\( 2\pi \)[/tex].
3. Identifying the Maximum:
- From the properties of the sine function, it is evident that the maximum value [tex]\( \sin(x) \)[/tex] can reach is 1, which it achieves at [tex]\( x = \frac{\pi}{2} + 2k\pi \)[/tex] for any integer [tex]\( k \)[/tex].
4. Selecting the Correct Answer:
- Given the choices [tex]\( -2\pi \)[/tex], [tex]\( -1 \)[/tex], [tex]\( 1 \)[/tex], and [tex]\( 2\pi \)[/tex], the maximum value of [tex]\( f(x) = \sin(x) \)[/tex] is indeed 1.
Therefore, the maximum value of [tex]\( f(x) = \sin(x) \)[/tex] is [tex]\( \boxed{1} \)[/tex].
