DefinitionSine and cosine Fourier transformations are used in constructing mathematical models to describe a real-valued function as an odd function or even function respectively. For real-valued odd functions, Fourier sine transforms break the function down into a sum of sine functions, while Fourier cosine transforms break real-valued even functions down into a sum of cosine functions. The type II cosine transform as well as the type III and type IV sine transforms have applications in solving boundary value problems, analyzing even or odd symmetric functions, and realizing the integral transforms of sampled functions. Sinusoidal series are applied to solve partial differential equations in mathematical physics and to analyze vibrations in signal processing, while Fourier sine and cosine transforms are applied to study heat conduction. In the same way, if f(x) is any function, then the Fourier sine transform is Fs(k)=∫0∞f(x)sin(kx)dx and the Fourier cosine transform is given by Fc(k)=∫0∞f(x)cos(kx)dx; where Fs(k) and Fc(k) refers to the frequency domain representations of the function.
