Waves are periodic function, as is the sine function.
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Sine graphs.
The obvious answer is the relationships between the sides and angles of triangles. Waves in the sea are an example of a sine wave. Tidal Experts and Meterologists alike use sine waves to help predict tides. Music will also emit waves that may often look like a sine wave and pure notes will look like sine or cosine waves. The speed of a swinging pendulum can be plotted as a sine wave as well as the sound of a tuning fork.
Electromagnetic radiation is propogated as sine waves.
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Sound waves are transmitted through a medium as variations in the pressure of the medium. If the variation is plotted as a function of distance (or time), they will generate a sine curve (the cosine curve is the same as a sine curve with a phase shift). In practise, the sine curve is damped: the amplitude (or height) of the oscillations gradually decrease over time or distance, because of attenuation.