The range of the function \( y = \sin x \) is the set of all possible output values (y-values) that the function can produce.
The sine function oscillates between -1 and 1 for all values of \( x \). This can be demonstrated through the properties of the sine function:
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Periodic Nature: The sine function is periodic with a period of \( 2\pi \). This means that it repeats its values every \( 2\pi \) units along the x-axis.
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Maximum and Minimum Values: The maximum value of \( \sin x \) occurs at \( \frac{\pi}{2} + 2k\pi \) (where \( k \) is any integer), and it is equal to 1. The minimum value occurs at \( \frac{3\pi}{2} + 2k\pi \), and it is equal to -1.
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Behavior of the Function: As \( x \) varies through all real numbers, \( \sin x \) will take every value from -1 to 1. There are infinitely many values of \( x \) that will give the same sine value since it is periodic.
Putting this all together, we conclude that the range of the function \( y = \sin x \) is:
\[ [-1, 1] \]
This means that \( \sin x \) can take any value between and including -1 and 1.
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