Linked Questions

2
votes
3answers
187 views

Is there any way to simplify the product of cosines? [duplicate]

I recently saw a problem: Estimate the following: $cos(\frac{\pi}{15})cos(\frac{2\pi}{15})\ldots cos(\frac{7\pi}{15})$, and the options were between different consecutive power of ten. How would I do ...
3
votes
2answers
161 views

Evaluate using complex numbers: $\prod^{n}_{k=1}\cos\left(\frac{k\pi}{m}\right)$, where $m=2n+1$ [duplicate]

Evaluate using complex numbers: $$\prod^{n}_{k=1}\cos\left(\frac{k\pi}{m}\right)$$ where $m=2n+1$. $\bf{My\; Try::}$ Let $\displaystyle P = \prod^{n}_{k=1}\cos\left(\frac{k\pi}{m}\right).$ Now let $\...
1
vote
0answers
75 views

Calculate ${2^{n + 1}}\prod\limits_{k = 1}^n {\cos \frac{{\pi k}}{{n + 1}}} $ [duplicate]

Calculate $${2^{n + 1}}\prod\limits_{k = 1}^n {\cos \frac{{\pi k}}{{n + 1}}} $$ It's task in theme "Eigenvectors and eigenvalues" (linear algebra book) I have no idea. Any advice? (How to solve it ...
0
votes
0answers
31 views

If $x=2\pi/2009$, then evaluate $\cos x \cos 2x \cos 3x \cdots \cos 1004x$ [duplicate]

I have been recently trying to solve a continued product of cosines: If $x=2\pi/2009$, then evaluate $$\cos x \cos 2x \cos 3x \cdots \cos 1004x$$ I have found formula for this which says product ...
31
votes
1answer
6k views

Fourier series of Log sine and Log cos

I saw the two identities $$ -\log(\sin(x))=\sum_{k=1}^\infty\frac{\cos(2kx)}{k}+\log(2) $$ and $$ -\log(\cos(x))=\sum_{k=1}^\infty(-1)^k\frac{\cos(2kx)}{k}+\log(2) $$ here: twist on classic log of ...
13
votes
1answer
4k views

Evaluation of a product of sines [duplicate]

Possible Duplicate: Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$ I am looking for a closed form for this product of sines: \begin{equation} \sin \left(\frac{\pi}{n}\...
4
votes
7answers
5k views

Prove that $\cos^2(\theta) + \cos^2(\theta +120^{\circ}) + \cos^2(\theta-120^{\circ})=3/2$

Prove that $$\cos^2(\theta) + \cos^2(\theta +120^{\circ}) + \cos^2(\theta-120^{\circ})=\frac{3}{2}$$ I thought of rewriting $$\cos^2(\theta +120^{\circ}) + \cos^2(\theta-120^{\circ})$$ as $$\cos^2(90^...
1
vote
2answers
166 views

Evaluating limit $\lim_{k\to \infty}\prod_{r=1}^k\cos{\left(\frac {x}{2^r}\right)}$

I stumbled across the following question which asked to evaluate... $$\lim_{k\to \infty}\prod_{r=1}^k\cos{\left(\frac {x}{2^r}\right)}$$ I at first tried writing few terms $$\cos{\left(\frac {x}{2}\...
2
votes
1answer
310 views

Prove that $\sum_{k=1}^{n} \frac1{\sin^2 \frac{\left( 2k-1\right)\pi}{4n+2}}=2n\left( n+1\right)$

Prove that $$\frac{1}{\sin^{2}\frac{\pi }{4k+2}}+\frac{1}{\sin^{2}\frac{3\pi }{4k+2}}+\frac{1}{\sin^{2}\frac{5\pi }{4k+2}}+\cdots+\frac{1}{\sin^{2}\frac{(2k-1)\pi }{4k+2}}=2k(k+1)$$
6
votes
0answers
198 views

Product of squared sines: $\prod_{k=1}^{n-1}\prod_{j=1}^{n-1}\left[ \sin^2\left(\frac{k\pi}{2 n}\right) +\sin^2\left(\frac{j\pi}{2 n}\right)\right]$

I have a double product $$a(n)=2^{2 (n-1)^2} \prod_{k=1}^{n-1} \prod_{j=1}^{n-1} \left[ \sin^2 \left(\frac{k\pi}{2 n}\right) +\sin^2 \left(\frac{j\pi}{2 n}\right) \right]$$ which always gives ...
3
votes
1answer
143 views

A golden trigonometric diophantine equation

After answering this question I reflected on the identity $$\cos\frac{\pi}{5}=\phi\cos\frac{\pi}{3}$$ and thought of looking for all the quadruplets of positive integers $(a,b,c,d)$ satisfying $$\cos \...
1
vote
0answers
56 views

Proving cosines product identity [duplicate]

Prove that $\cos\left({\pi \over 11}\right)\cdot\cos\left({2\pi \over 11}\right)\cdot\cos\left({3\pi \over 11}\right)\cdot\cos\left({4\pi \over 11}\right)\cdot\cos\left({5\pi \over 11}\right)={1 \...
1
vote
0answers
42 views

Evaluate $ \cos(x) \cos(2x) \cos(3x) … \cos(999x) $ where $x = \frac{2 \pi}{ 1999 }$. [duplicate]

Evaluate $$ \cos(x) \cos(2x) \cos(3x) ... \cos(999x) $$ where $x = \frac{2 \pi}{ 1999 }$. Attempt: Let $A = \cos(x) \cos(2x) \cos(3x) ... \cos(999x)$, $B = \sin(x) \sin(2x) \sin(3x) ... \sin(999x)$...