Yes: they represent the same thing.
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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 real life example of the sine function could be a ferris wheel. People board the ride at the ground (sinusoidal axis) and the highest and lowest heights you reach on the ride would be the amplitudes of the graph.
Its periodicity and amplitude remain the same.
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.