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  "body": "\n## Summary\n\nThe sine and cosine transforms are the real forms of the [[Fourier transform|DPVM34ADZV]] for functions on the half-line $[0, \\infty)$.\nThe cosine transform integrates $f$ against $\\cos(2\\pi x \\xi)$, and the sine transform integrates it against $\\sin(2\\pi x \\xi)$.\nThey are the Fourier transforms of the even and of the odd extension of $f$, up to constant factors.\nThey are used to solve boundary value problems on a half-line, where the boundary condition picks the even or the odd extension.\nSee the [[definition|Q4HR7AF2R9]].\n\n## Explanations\n\n### L1 human\n\nA signal that starts at time $0$ can be mirrored to negative times in two ways: as an even copy or as an odd copy.\nThe mirrored signal has a Fourier transform made only of cosines in the even case and only of sines in the odd case.\nSo one real transform suffices instead of a complex one.\nFor $f(x) = e^{-x}$ on $[0, \\infty)$, the cosine transform is $1/(1 + 4\\pi^2 \\xi^2)$.\nThese transforms are the continuous analogues of the [[sine and cosine series|Y6EPBG6S8G]] of a function on an interval.\n",
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