Using the formulae for $\cos (A \pm B)$, prove that $\cos \left( r+\frac{1}{2} ight) heta - \cos

Question

Using the formulae for $\cos (A \pm B)$, prove that $\cos \left( r+\frac{1}{2} ight)	heta - \cos \left( r-\frac{1}{2} ight)	heta = -2\sin r	heta \sin \frac{1}{2}	heta$. Hence find the formula for $\sum\limits_{r=1}^{n}{\sin r	heta}$ in terms of $2\sin \frac{1}{2}	heta$, $\cos \left( n+\frac{1}{2} ight)	heta$, and $\cos \frac{1}{2}	heta$.

Accepted Answer

(b) $\sum_{r=1}^{n} \sin(r\theta) = -\frac{1}{2\sin\frac{\theta}{2}}\left[\cos\left(n + \frac{1}{2}\right)\theta - \cos\frac{\theta}{2}\right]$

Sigma Notation By Substitution Q5

Sigma Notation By Substitution Q5

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