To determine the range of the function [tex]\( y = \sin \theta \)[/tex], we need to find the set of all possible [tex]\( y \)[/tex]-values that the sine function can produce as [tex]\( \theta \)[/tex] varies over all real numbers.
1. Understanding the sine function:
The sine function, [tex]\( \sin \theta \)[/tex], is a periodic function with a period of [tex]\( 2\pi \)[/tex]. This means it repeats its values every [tex]\( 2\pi \)[/tex] units.
2. Range of values:
The sine function oscillates up and down continuously and smoothly between its maximum and minimum values. The maximum value of [tex]\( \sin \theta \)[/tex] is 1, and the minimum value is -1.
3. Specific Values:
- At [tex]\( \theta = 0 \)[/tex]: [tex]\( \sin(0) = 0 \)[/tex]
- At [tex]\( \theta = \pi/2 \)[/tex]: [tex]\( \sin(\pi/2) = 1 \)[/tex]
- At [tex]\( \theta = \pi \)[/tex]: [tex]\( \sin(\pi) = 0 \)[/tex]
- At [tex]\( \theta = 3\pi/2 \)[/tex]: [tex]\( \sin(3\pi/2) = -1 \)[/tex]
- The values repeat every [tex]\( 2\pi \)[/tex].
4. Conclusion:
Therefore, the sine function [tex]\( \sin \theta \)[/tex] takes on all values from [tex]\( -1 \)[/tex] to [tex]\( 1 \)[/tex], inclusive. Thus, the range of [tex]\( y = \sin \theta \)[/tex] is [tex]\( [-1, 1] \)[/tex].
The correct answer is:
A. [tex]\([-1, 1]\)[/tex]
