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  "body": "\n## Summary\n\nA trigonometric polynomial is a finite linear combination of the functions $e^{inx}$, $n \\in \\Z$, or equivalently of $1$, $\\cos nx$ and $\\sin nx$.\nTrigonometric polynomials are the partial sums of [[Fourier series|G12KXDHPPS]].\nThey are dense in the continuous $2\\pi$-periodic functions, so they play for periodic functions the role that ordinary polynomials play on a closed interval.\n\n## Explanations\n\n### L1 human\n\nA trigonometric polynomial is a sum of finitely many pure waves, each with a whole-number frequency.\nIts graph repeats itself after every interval of length $2\\pi$.\nAn example is $T(x) = 1 + 2\\cos x - 3 \\sin 2x$, which has degree $2$, the largest frequency present.\nWriting $z = e^{ix}$ turns it into a sum of powers $z^n$ with $n$ positive or negative, which explains the word polynomial.\n",
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