Trigonometry
Product of cosines as power of 2
MJAT_TS3_P2
Grade 12
Question:
If $\displaystyle\cos\frac{2\pi}{2019}\cdot\cos\frac{4\pi}{2019}\cdot\cos\frac{6\pi}{2019}\cdots\cos\frac{2018\pi}{2019} = 2^{-k}$ where $k=\dfrac{p}{q}$, with $p,q\in\mathbb{N}$ coprime, then:
A) $(p+q)$ is a 4-digit number
B) Sum of the digits in $(p+q)$ is $2$
C) $(p+q)$ is a 3-digit number
D) Number of divisors of $p$ is $8$
Step-by-Step Solution
Key Concept: Use the identity: $\prod_{k=1}^{n-1}\sin\frac{k\pi}{n}=\frac{n}{2^{n-1}}$ and double-angle formula to relate the product of cosines. For $n=2019$ (prime), $\prod_{k=1}^{1009}\cos\frac{2k\pi}{2019}=\frac{1}{2^{1009}}$.
$k=1009$, $p=1009$, $q=1$, $p+q=1010$. A ✓, B ✓. Answer: A, B.
Correct Answer: AB