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I want to express $$f(x) = \sin^3x$$ in terms of its fourier series using the fourier series of $sin2x$ and $cos^2 x$ between the interval $[-\pi, \pi]$ for a period of $2\pi$. Respectively here is what I found:

$$\sin2x = \sum^{\infty}_1b_n \sin(nx)$$

$$\cos^2x = \frac{a_0}{2} + \sum^{\infty}_1a_n cos(nx)$$

But from what I have searched online, $sin^3x$ requires an expression in terms of $sin3x$, so I'm unable to use this fact with my requirements of $cosx$ and sin$2x$

user577215664
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oldselflearner1959
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  • It should be Fourier, with a capital letter. 2. $\sin^3 x$ is an odd function, hence its Fourier series only has sine terms 3. $$\left(\frac{e^{ix}-e^{-ix}}{2i}\right)^3 = \ldots = \frac{3}{4}\sin(x)-\frac{1}{4}\sin(3x).$$
    • – Jack D'Aurizio Mar 12 '18 at 19:21
  • Possible duplicate of Is $\sin^3 x=\frac{3}{4}\sin x - \frac{1}{4}\sin 3x$? – Leucippus Mar 13 '18 at 02:46