What is the range of [tex]f(x)=\sin (x)[/tex]?
A. the set of all real numbers [tex]-2 \pi \leq y \leq 2 \pi[/tex]
B. the set of all real numbers [tex]-1 \leq y \leq 1[/tex]
C. the set of all real numbers [tex]0 \leq y \leq 2 \pi[/tex]
D. the set of all real numbers
Answer (2)
The range of the function f ( x ) = sin ( x ) is the set of all real numbers − 1 ≤ y ≤ 1 . This means that the sine function outputs values between -1 and 1. Therefore, the correct choice is B .
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The sine function, sin ( x ) , represents the y-coordinate on the unit circle.
The y-coordinate varies between -1 and 1 as the point moves around the unit circle.
The maximum value of sin ( x ) is 1, and the minimum value is -1.
Therefore, the range of f ( x ) = sin ( x ) is − 1 ≤ y ≤ 1 .
Explanation
Understanding the Range We are asked to find the range of the function f ( x ) = s in ( x ) . The range of a function is the set of all possible output values (y-values) that the function can take.
Sine Function and the Unit Circle Let's consider the unit circle. The sine function, s in ( x ) , represents the y-coordinate of a point on the unit circle, where x is the angle formed with the positive x-axis. As the point moves around the unit circle, the y-coordinate varies between -1 and 1.
Maximum and Minimum Values The maximum value of s in ( x ) is 1, which occurs at angles like 2 π , 2 5 π , etc. The minimum value of s in ( x ) is -1, which occurs at angles like 2 3 π , 2 7 π , etc. Therefore, the sine function oscillates between -1 and 1.
Determining the Range Thus, the range of f ( x ) = s in ( x ) is the set of all real numbers y such that − 1 l e q y l e q 1 .
Examples
Understanding the range of trigonometric functions like sine is crucial in many real-world applications. For example, in electrical engineering, alternating current (AC) voltage and current are often modeled using sinusoidal functions. Knowing that the sine function's range is [-1, 1] helps engineers determine the maximum and minimum voltage or current values in a circuit, which is essential for designing safe and efficient electrical systems. Similarly, in acoustics, sound waves can be modeled using sine functions, and the range helps determine the maximum and minimum sound pressure levels.
