To determine whether the statement [tex]\(\sin(-x) = -\sin(x)\)[/tex] is true for all values of [tex]\(x\)[/tex], let's analyze the properties of the sine function.
1. Understanding the Sine Function:
The sine function, [tex]\(\sin(x)\)[/tex], is one of the fundamental trigonometric functions, which is periodic with a period of [tex]\(2\pi\)[/tex] radians. This function has specific properties that help us understand its behavior.
2. Odd Function Property:
A function [tex]\(f(x)\)[/tex] is called an odd function if it satisfies the condition [tex]\(f(-x) = -f(x)\)[/tex] for all [tex]\(x\)[/tex] in its domain.
3. Checking the Given Statement:
To determine if [tex]\(\sin(-x) = -\sin(x)\)[/tex] is true for all [tex]\(x\)[/tex], we check if the sine function satisfies the property of odd functions.
4. Evaluation:
- For [tex]\(x = 0\)[/tex], [tex]\(\sin(0) = 0\)[/tex]. Therefore, [tex]\(\sin(-0) = \sin(0) = 0\)[/tex] and [tex]\(-\sin(0) = 0\)[/tex]. Thus, [tex]\(\sin(-0) = -\sin(0)\)[/tex].
- For [tex]\(x = \pi/2\)[/tex], [tex]\(\sin(\pi/2) = 1\)[/tex]. Therefore, [tex]\(\sin(-\pi/2) = -1\)[/tex] and [tex]\(-\sin(\pi/2) = -1\)[/tex]. Thus, [tex]\(\sin(-\pi/2) = -\sin(\pi/2)\)[/tex].
- For [tex]\(x = \pi\)[/tex], [tex]\(\sin(\pi) = 0\)[/tex]. Therefore, [tex]\(\sin(-\pi) = 0\)[/tex] and [tex]\(-\sin(\pi) = 0\)[/tex]. Thus, [tex]\(\sin(-\pi) = -\sin(\pi)\)[/tex].
- For any general [tex]\(x\)[/tex], consider the sine curve symmetry about the origin. The sine of a positive angle and the sine of its corresponding negative angle are the same in magnitude but opposite in sign due to the wave nature of the sine function.
5. Conclusion:
Since our evaluations confirm that for any angle [tex]\(x\)[/tex], [tex]\(\sin(-x) = -\sin(x)\)[/tex] holds true, we see that the sine function behaves as an odd function.
Therefore, the statement [tex]\(\sin(-x) = -\sin(x)\)[/tex] is true for all values of [tex]\(x\)[/tex].
The correct answer is:
A. True
