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The cosine is a projection of the complex number exp(−ix), which lies on the unit circle in the complex plane, to the real axis on the complex plane. The behavior of the cosine function can be studied in the following interactive figure, where you can drag the point x on the real axis and observe the behaviour of the complex number exp(−ix) and the value of cosine(x).

Cosine Unit Circle on the Complex Plane

read more about cosine on Bookofproofs