Sine is an odd function

This trigonometric identity states that if you take the sine of a negative angle, it is the same as taking the negative of the sine of that angle's positive counterpart.

This property classifies sine as an odd function because it satisfies the general definition of odd functions, where \(\htmlId{tooltip-function}{f}(-\htmlId{tooltip-var}{x}) = -\htmlId{tooltip-fOfX}{f(x)}\) for any value of \(\htmlId{tooltip-var}{x}\). Visually, this means the graph of the sine function is symmetric about the origin, reflecting across both the \( \htmlId{tooltip-var}{x} \)-axis and \( \htmlId{tooltip-2var}{y} \)-axis. The odd nature of the sine function is fundamental in trigonometry and is used to analyze and solve problems involving periodic and harmonic motions.

1. Consider the definition of an odd function:
   \[-\htmlId{tooltip-fOfX}{f(x)} = \htmlId{tooltip-function}{f}(-\htmlId{tooltip-var}{x})\]
   To show that the sine function is odd, we need to demonstrate that this property holds for \(\htmlId{tooltip-sine}{\sin}(\htmlId{tooltip-var}{x})\)
2. The sine function can be understood geometrically using the unit circle, where the sine of an angle \(\htmlId{tooltip-var}{x}\) (measured in radians (\( \htmlId{tooltip-u-radians}{rad} \))) represents the \( \htmlId{tooltip-2var}{y} \)-coordinate of the point on the unit circle at that angle from the positive \( \htmlId{tooltip-var}{x} \)-axis.
3. When we consider a negative angle \(-\( \htmlId{tooltip-var}{x} \)), /we're looking at an angle measured in the opposite direction (clockwise). This takes us to a point on the unit circle that is directly below the \( \htmlId{tooltip-var}{x} \)-axis if the point for \(\htmlId{tooltip-var}{x}\) is directly above the \( \htmlId{tooltip-var}{x} \)-axis. This means if the point for \(\htmlId{tooltip-var}{x}\) has coordinates \((\htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-var}{x}), \htmlId{tooltip-sine}{\sin}(\htmlId{tooltip-var}{x}))\), then the point for \(-\htmlId{tooltip-var}{x}\) has coordinates \((\htmlId{tooltip-cosine}{\cos}(-\htmlId{tooltip-var}{x}), \htmlId{tooltip-sine}{\sin}(-\htmlId{tooltip-var}{x}))\).
4. We know that cosine is an even function consider:
   \[\htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-angle}{\theta}) = \htmlId{tooltip-cosine}{\cos}(-\htmlId{tooltip-angle}{\theta})\] This is because the cosine function measures the x-coordinate of a point on the unit circle, which is unaffected by flipping the angle above or below the \( \htmlId{tooltip-var}{x} \)-axis.
5. However, flipping the angle to \(-\htmlId{tooltip-var}{x}\) changes the sign of the \( \htmlId{tooltip-2var}{y} \)-coordinate (because we're moving to the opposite side of the \( \htmlId{tooltip-var}{x} \)-axis), and therefore we have:
   \[\htmlId{tooltip-sine}{\sin}(-\htmlId{tooltip-angle}{\theta}) = -\htmlId{tooltip-sine}{\sin}(\htmlId{tooltip-angle}{\theta})\] This shows that taking the sine of a negative angle gives the negative of the sine of the positive angle.

If we take \(\htmlId{tooltip-var}{x} = \frac{\htmlId{tooltip-pi}{\pi}}{6}\). We can easily see that:

\[\htmlId{tooltip-sine}{\sin}(-\frac{\htmlId{tooltip-pi}{\pi}}{6}) = -\htmlId{tooltip-sine}{\sin}(\frac{\htmlId{tooltip-pi}{\pi}}{6})\]