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How did Joseph Fourier use sine and cosine functions to represent periodic patterns?

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Diagram of a right triangle labeled with sides a, b, and hypotenuse c, alongside formulas for the six trigonometric functions-sine, cosine, tangent, cosecant, secant, and cotangent-each defined by ratios of the triangle's sides. — trigonometry Based on the definitions, various simple relationships exist among the functions. For example, csc A = 1/sin A, sec A = 1/cos A, cot A = 1/tan A, and tan A = sin A/cos A.

Joseph Fourier revolutionized mathematics by demonstrating that almost any periodic function can be represented as an infinite sum of sine and cosine functions, a concept now known as the Fourier series.12

In his work, particularly regarding heat conduction, Fourier showed that complex periodic patterns—such as those found in sound waves, tides, or vibrations—could be broken down into simpler components. In this series, one term corresponds to the fundamental period of the function, while the remaining terms, called harmonics, have periods that are integral submultiples of that fundamental period.342

By summing these trigonometric terms, Fourier provided a method to approximate complex curves with increasing accuracy as more terms are added. This discovery transformed mathematical physics and remains a fundamental tool in modern engineering, acoustics, and optics.123