Question 0511

Sigma Notation

2011 Paper 1 Question 6 Variant

Question

(a)

Using the formulae for

{\cos(A \pm B),}

prove that

\cos {\textstyle r} \theta - \cos {\textstyle (r + 1)} = 2 \sin { \textstyle \frac{2r+1}{2}} \theta \sin {\textstyle \frac{1}{2} \theta}.

[2]

(b)

Hence find a formula for

\sum_{r=1}^n \sin  { \textstyle \frac{2r+1}{2}} \theta

in terms of

{\cos (n+1) \theta,}

{\cos \theta}

and

{\sin \frac{1}{2}\theta.}

[3]