Question

which trigonometric function is an odd function and why?
the sine function is odd because it is represented by the y-coordinate of the points on the unit circle, and therefore sin(x) = sin(-x).
the cosine function is odd because it is represented by the x-coordinate of the points on the unit circle, and therefore cos(-x) = -cos(x).
the cosine function is odd because it is represented by the x-coordinate of the points on the unit circle, and therefore cos(x) = cos(-x).
the sine function is odd because it is represented by the y-coordinate of the points on the unit circle, and therefore sin(-x) = -sin(x).

Explanation:

An odd function satisfies \(f(-x)=-f(x)\). For the sine function, using the unit - circle definition (where \(y = \sin(x)\)), if we consider the angle \(-x\), the \(y\) - coordinate of the point on the unit circle corresponding to \(-x\) is the negative of the \(y\) - coordinate of the point corresponding to \(x\). So, \(\sin(-x)=-\sin(x)\).

The cosine function is an even function since \(\cos(-x)=\cos(x)\) (using the \(x\) - coordinate of the unit - circle points). The first option has the wrong equation (\(\sin(x)=\sin(-x)\) is incorrect for an odd function), the second and third options are wrong because the cosine function is not odd.

Answer:

The sine function is odd because it is represented by the \(y\) - coordinate of the points on the unit circle, and therefore \(\sin(-x)=-\sin(x)\).