Given that $\mathrm{f}\left( r ight)=\cos 2r heta $, write $\mathrm{f}\left( r ight)-\mathrm{f}

Question

Given that $\mathrm{f}\left( r ight)=\cos 2r	heta $, write $\mathrm{f}\left( r ight)-\mathrm{f}\left( r+1 ight)$ as a product of two sine functions. Use your result to find the sum of the first $n$ terms of the series $\sin 3	heta +\sin 5	heta +\sin 7	heta +...$

Accepted Answer

$f(r)-f(r+1)=2\sin((2r+1)\theta)\sin\theta$. For $\sin\theta\ne0$, $\displaystyle\sum_{r=1}^{n}\sin((2r+1)\theta)=\frac{\cos2\theta-\cos(2(n+1)\theta)}{2\sin\theta}$. For $\theta\in\pi\mathbb Z$, the sum is $0$.

HCI Sequence and Series Tutorial Q5

HCI Sequence and Series Tutorial Q5

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