Math Problem 7

yes, I’m aware I haven’t posted in a while, and in my defence you should’ve known I’m a slacker. Anyway, as I don’t want to deal with real life shit, I’m using this as a platform to temporarily escape real life like a coward. While I won’t go into details about what shit am I currently facing right now, let’s just say I have real homework and revision yet to be finished and my social life is fucked.

(a) Let $\theta\in\mathbb R$. Using mathematical induction, prove that
$\displaystyle\sin\theta\sum_{k=1}^n \sin(2k\theta)=\sin(n\theta)\sin((n+1)\theta)$
for all positive integers $n$.

(b) Using part (a), find rational numbers $a$ and $b$ such that
$\displaystyle\sum_{k=1}^{111} \sin\frac{k\pi}{11}\cos\frac{k\pi}{11}=a\sin(b\pi)$, where $0<b<\tfrac12$.

($5 + 3$ marks)

the following question was completed on the 16th of Feb, 2026

Suggested Solution

fml ngl im cooked