To understand which trigonometric function is odd, we refer to their fundamental definitions and properties.
An odd function [tex]\( f(x) \)[/tex] is one that satisfies the condition:
[tex]\[ f(-x) = -f(x) \][/tex]
1. Sine Function:
The sine function is defined as the [tex]\( y \)[/tex]-coordinate of the point on the unit circle corresponding to an angle [tex]\( x \)[/tex]. Therefore, for the sine function:
[tex]\[ \sin(-x) = -\sin(x) \][/tex]
This confirms that the sine function is indeed an odd function because it satisfies the condition for oddness.
2. Cosine Function:
The cosine function is defined as the [tex]\( x \)[/tex]-coordinate of the point on the unit circle corresponding to an angle [tex]\( x \)[/tex]. Therefore, for the cosine function:
[tex]\[ \cos(-x) = \cos(x) \][/tex]
This confirms that the cosine function is an even function, not an odd function because it does not satisfy the condition for oddness.
Hence, we conclude that:
- The sine function is odd because [tex]\(\sin(-x) = -\sin(x) \)[/tex].
- The cosine function is not odd because [tex]\(\cos(-x) = \cos(x) \)[/tex].
Given the options:
- The correct statement is: The sine function is odd because it is represented by the [tex]\( y \)[/tex]-coordinate of the points on the unit circle, and therefore [tex]\(\sin(-x) = -\sin(x)\)[/tex].
This reasoning aligns with our understanding and the properties of trigonometric functions.
