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 (1)
The range of a function is the set of all possible output values.
The sine function, sin ( x ) , represents the y-coordinate on the unit circle.
The y-coordinate on the unit circle varies between -1 and 1, inclusive.
The range of f ( x ) = sin ( x ) is ā 1 ā¤ y ā¤ 1 ā .
Explanation
Understanding the Range The problem asks for 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 produce. To find the range of the sine function, we need to determine the minimum and maximum values that s in ( x ) can take.
Sine Function and the Unit Circle Recall the unit circle definition of s in ( x ) . For any angle x , s in ( x ) corresponds to the y-coordinate of the point on the unit circle that corresponds to the angle x . The unit circle has a radius of 1, so the y-coordinate can never be greater than 1 or less than -1.
Determining Minimum and Maximum Values The maximum value of s in ( x ) is 1, which occurs when x = f r a c p i 2 + 2 kĻ for any integer k . The minimum value of s in ( x ) is -1, which occurs when x = f r a c 3 p i 2 + 2 kĻ for any integer k . Therefore, the range of s in ( x ) is the set of all real numbers between -1 and 1, inclusive.
Final Answer The range of f ( x ) = s in ( x ) is the set of all real numbers ā 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 between -1 and 1 helps engineers determine the maximum and minimum voltage or current values in an AC 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.
