If you don't understand the mathematical term "orthogonalized", I'm not sure that any explanation is going to help you much. It basically means a set of wave functions which are independent... that is, the value of one does not depend on the value of any of the others.
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The differential of the sine function is the cosine function while the differential of the cosine function is the negative of the sine function.
By its very mane, a sinusoidal wave refers to a sine function. The cosine function is simply the sine function that is phase-shifted.
A sine wave is a periodic function and, by suitably adjusting the argument of the sine function, can be made to fit a wide functions with different frequencies.
A sine wave is the graph of y = sin(x). It demonstrates to cyclic nature of the sine function.
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.