What is the maximum of [tex]f(x)=\sin (x)[/tex]?
A. [tex]-2 \pi[/tex]
B. -1
C. 1
D. [tex]2 \pi[/tex]
Answer (1)
The problem asks for the maximum value of the sine function, f ( x ) = sin ( x ) .
The range of the sine function is − 1 ≤ sin ( x ) ≤ 1 .
The maximum value of the sine function is therefore 1.
The final answer is 1 .
Explanation
Problem Analysis We are asked to find the maximum value of the function f ( x ) = sin ( x ) and choose the correct answer from the given options.
Understanding the Sine Function The sine function, f ( x ) = sin ( x ) , oscillates between -1 and 1. This means that for any value of x , the value of sin ( x ) will always be greater than or equal to -1 and less than or equal to 1. In mathematical terms, the range of the sine function is − 1 ≤ sin ( x ) ≤ 1 .
Determining the Maximum Value Therefore, the maximum value of the sine function is 1.
Comparing with Options Now, let's compare this to the given options:
− 2 π is approximately -6.28, which is less than -1.
− 1 is the minimum value of the sine function.
1 is the maximum value of the sine function.
2 π is approximately 6.28, which is greater than 1.
Final Answer The maximum value of f ( x ) = sin ( x ) is 1.
Examples
The sine function is used to model many real-world phenomena that exhibit periodic behavior, such as the oscillation of a pendulum, the propagation of light waves, and the variation of alternating current (AC) voltage. For example, in electrical engineering, the voltage V ( t ) in an AC circuit can be modeled as V ( t ) = V 0 sin ( 2 π f t ) , where V 0 is the peak voltage and f is the frequency. Knowing that the maximum value of the sine function is 1 helps engineers determine the peak voltage in the circuit, which is crucial for designing and analyzing electrical systems.
