To determine the maximum and minimum values of the sine function using the unit circle, we need to analyze the sine function's behavior on the unit circle.
1. Understanding the Unit Circle:
- A unit circle is a circle with a radius of 1, centered at the origin (0, 0) in the coordinate plane.
- Any point [tex]\( P \)[/tex] on the unit circle can be represented as [tex]\( (x, y) \)[/tex], and it satisfies the equation [tex]\( x^2 + y^2 = 1 \)[/tex].
2. Sine Function on the Unit Circle:
- The sine of an angle [tex]\( \theta \)[/tex] in the unit circle is defined as the y-coordinate of the point [tex]\( P \)[/tex] associated with that angle [tex]\( \theta \)[/tex].
- This means that the value of the sine function [tex]\( \sin(\theta) \)[/tex] corresponds to the vertical distance from the x-axis to the point [tex]\( P \)[/tex].
3. Maximum Value:
- As you move around the unit circle, the y-coordinate (or the sine value) reaches its highest value at the topmost point of the circle.
- The topmost point of the unit circle is (0, 1).
- Hence, the maximum value of the sine function is [tex]\( 1 \)[/tex].
4. Minimum Value:
- Similarly, the y-coordinate (or the sine value) reaches its lowest value at the bottommost point of the circle.
- The bottommost point of the unit circle is (0, -1).
- Hence, the minimum value of the sine function is [tex]\( -1 \)[/tex].
From this analysis, we see that the maximum value of the sine function on the unit circle is [tex]\( 1 \)[/tex] and the minimum value is [tex]\( -1 \)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{(-1, 1)}
\][/tex]
