Yes
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Yes. Along with the tangent function, sine is an odd function. Cosine, however, is an even function.
Yes.
A basic wave function is a sine or cosine function whose amplitude may have a value other than 1. The cosine function is an even function because it is symmetrical about the y-axis. That is, f(-x) = f(x) for all x. The sine function is an odd function because it is antisymmetrical about the y-axis. That is, f(-x) = -f(x) for all x.
The differential of the sine function is the cosine function while the differential of the cosine function is the negative of the sine function.
Waves are periodic function, as is the sine function.