What is the Product of a Lot of Cosines?

[caffeinemachine]

Hello MHB. I need help with the follwoing:

Evaluate $\displaystyle P=\prod_{j=1}^{1004}\cos\left(\frac{2\pi j}{2009}\right)$.

I think it would be easier to first evaluate $\displaystyle Q=\prod_{j=1}^{2008}\cos\left(\frac{2\pi j}{2009}\right)$. Knowing the value of $Q$ we can easily find the value of $P$.

Can anybody help?

 

[I like Serena]

Hello MHB. I need help with the follwoing:

Evaluate $\displaystyle P=\prod_{j=1}^{1004}\cos\left(\frac{2\pi j}{2009}\right)$.

I think it would be easier to first evaluate $\displaystyle Q=\prod_{j=1}^{2008}\cos\left(\frac{2\pi j}{2009}\right)$. Knowing the value of $Q$ we can easily find the value of $P$.

Can anybody help?

Hi caffeinemachine! :)

When you write $$\cos\left(\frac{2\pi j}{2009}\right) = \frac 1 2 \left(e^{2\pi i j/2009} + e^{-2\pi i j/2009}\right)$$, you can simplify P into a sum that turns out to be a geometric series with almost everything canceling...

 

[caffeinemachine]

Hi caffeinemachine! :)

When you write $$\cos\left(\frac{2\pi j}{2009}\right) = \frac 1 2 \left(e^{2\pi i j/2009} + e^{-2\pi i j/2009}\right)$$, you can simplify P into a sum that turns out to be a geometric series with almost everything canceling...

Thank you I Like Serena. This solved the problem. :)
Wonder how can we do it without invoking Euler's Formula.

 

[anemone]

Hello MHB. I need help with the follwoing:

Evaluate $\displaystyle P=\prod_{j=1}^{1004}\cos\left(\frac{2\pi j}{2009}\right)$.

I think it would be easier to first evaluate $\displaystyle Q=\prod_{j=1}^{2008}\cos\left(\frac{2\pi j}{2009}\right)$. Knowing the value of $Q$ we can easily find the value of $P$.

Can anybody help?

Hi caffeinemachine,

I like Serena's proposed method is great, of course, but I've another method in mind that I want to share with you.

$$P=\prod_{j=1}^{1004}\cos\left(\frac{2\pi j}{2009}\right)$$

$$P=\cos\left(\frac{2\pi }{2009}\right)\cos\left(\frac{4\pi }{2009}\right)\cos\left(\frac{6\pi }{2009}\right)\cdots\cos\left(\frac{2008\pi }{2009}\right)$$

Let

$$Q=\prod_{j=1}^{1004}\sin\left(\frac{2\pi j}{2009}\right)$$

$$Q=\sin\left(\frac{2\pi }{2009}\right)\sin\left(\frac{4\pi }{2009}\right)\sin\left(\frac{6\pi }{2009}\right)\cdots\sin\left(\frac{2008\pi }{2009}\right)$$

Hence

$$PQ=\frac{1}{2^{1004}}\sin\left(\frac{4\pi }{2009}\right)\sin\left(\frac{8\pi }{2009}\right)\sin\left(\frac{12\pi }{2009}\right)\cdots\sin\left(\frac{4016\pi }{2009}\right)$$

$$PQ=\frac{1}{2^{1004}}\small\left(\sin\left(\frac{4\pi }{2009}\right)\sin\left(\frac{8\pi }{2009}\right)\sin\left(\frac{12\pi }{2009}\right)\cdots\sin\left(\frac{2008\pi }{2009}\right)\right)\left(\sin\left(\frac{2012\pi }{2009}\right)\sin\left(\frac{2016\pi }{2009}\right)\sin\left(\frac{2020\pi }{2009}\right)\cdots\sin\left(\frac{4016\pi }{2009}\right)\right)$$

Note that

$$\sin\left(\frac{2012\pi }{2009}\right)=\sin\left(2\pi-\frac{2006\pi }{2009}\right)=-\sin\left(\frac{2006\pi }{2009}\right)$$

and

$$\sin\left(\frac{2016\pi }{2009}\right)=\sin\left(2\pi-\frac{2002\pi }{2009}\right)=-\sin\left(\frac{2002\pi }{2009}\right)$$ and so on and so forth,

it's obvious that

$$PQ=\frac{1}{2^{1004}}(-1)^{502}\sin\left(\frac{2\pi }{2009}\right)\sin\left(\frac{4\pi }{2009}\right)\sin\left(\frac{6\pi }{2009}\right)\cdots\sin\left(\frac{2008\pi }{2009}\right)$$

$$PQ=\frac{1}{2^{1004}}Q$$

$$\therefore P=\frac{1}{2^{1004}}$$

P.S. You might want to read this thread as well!http://mathhelpboards.com/trigonometry-12/simplify-cos-cos-2a-cos-3a-cos-999a-if-%3D-2pi-1999-a-253.html

 

[caffeinemachine]

Hi caffeinemachine,

I like Serena's proposed method is great, of course, but I've another method in mind that I want to share with you.

$$P=\prod_{j=1}^{1004}\cos\left(\frac{2\pi j}{2009}\right)$$

$$P=\cos\left(\frac{2\pi }{2009}\right)\cos\left(\frac{4\pi }{2009}\right)\cos\left(\frac{6\pi }{2009}\right)\cdots\cos\left(\frac{2008\pi }{2009}\right)$$

Let

$$Q=\prod_{j=1}^{1004}\sin\left(\frac{2\pi j}{2009}\right)$$

$$Q=\sin\left(\frac{2\pi }{2009}\right)\sin\left(\frac{4\pi }{2009}\right)\sin\left(\frac{6\pi }{2009}\right)\cdots\sin\left(\frac{2008\pi }{2009}\right)$$

Hence

$$PQ=\frac{1}{2^{1004}}\sin\left(\frac{4\pi }{2009}\right)\sin\left(\frac{8\pi }{2009}\right)\sin\left(\frac{12\pi }{2009}\right)\cdots\sin\left(\frac{4016\pi }{2009}\right)$$

$$PQ=\frac{1}{2^{1004}}\small\left(\sin\left(\frac{4\pi }{2009}\right)\sin\left(\frac{8\pi }{2009}\right)\sin\left(\frac{12\pi }{2009}\right)\cdots\sin\left(\frac{2008\pi }{2009}\right)\right)\left(\sin\left(\frac{2012\pi }{2009}\right)\sin\left(\frac{2016\pi }{2009}\right)\sin\left(\frac{2020\pi }{2009}\right)\cdots\sin\left(\frac{4016\pi }{2009}\right)\right)$$

Note that

$$\sin\left(\frac{2012\pi }{2009}\right)=\sin\left(2\pi-\frac{2006\pi }{2009}\right)=-\sin\left(\frac{2006\pi }{2009}\right)$$

and

$$\sin\left(\frac{2016\pi }{2009}\right)=\sin\left(2\pi-\frac{2002\pi }{2009}\right)=-\sin\left(\frac{2002\pi }{2009}\right)$$ and so on and so forth,

it's obvious that

$$PQ=\frac{1}{2^{1004}}(-1)^{502}\sin\left(\frac{2\pi }{2009}\right)\sin\left(\frac{4\pi }{2009}\right)\sin\left(\frac{6\pi }{2009}\right)\cdots\sin\left(\frac{2008\pi }{2009}\right)$$

$$PQ=\frac{1}{2^{1004}}Q$$

$$\therefore P=\frac{1}{2^{1004}}$$

P.S. You might want to read this thread as well!http://mathhelpboards.com/trigonometry-12/simplify-cos-cos-2a-cos-3a-cos-999a-if-%3D-2pi-1999-a-253.html

That's masterful. And very nicely typed in LaTeX. I was wondering why anemone has not yet answered this question. :)

 

[kaliprasad]

Hi caffeinemachine,

I like Serena's proposed method is great, of course, but I've another method in mind that I want to share with you.

$$P=\prod_{j=1}^{1004}\cos\left(\frac{2\pi j}{2009}\right)$$

$$P=\cos\left(\frac{2\pi }{2009}\right)\cos\left(\frac{4\pi }{2009}\right)\cos\left(\frac{6\pi }{2009}\right)\cdots\cos\left(\frac{2008\pi }{2009}\right)$$

Let

$$Q=\prod_{j=1}^{1004}\sin\left(\frac{2\pi j}{2009}\right)$$

$$Q=\sin\left(\frac{2\pi }{2009}\right)\sin\left(\frac{4\pi }{2009}\right)\sin\left(\frac{6\pi }{2009}\right)\cdots\sin\left(\frac{2008\pi }{2009}\right)$$

Hence

$$PQ=\frac{1}{2^{1004}}\sin\left(\frac{4\pi }{2009}\right)\sin\left(\frac{8\pi }{2009}\right)\sin\left(\frac{12\pi }{2009}\right)\cdots\sin\left(\frac{4016\pi }{2009}\right)$$

$$PQ=\frac{1}{2^{1004}}\small\left(\sin\left(\frac{4\pi }{2009}\right)\sin\left(\frac{8\pi }{2009}\right)\sin\left(\frac{12\pi }{2009}\right)\cdots\sin\left(\frac{2008\pi }{2009}\right)\right)\left(\sin\left(\frac{2012\pi }{2009}\right)\sin\left(\frac{2016\pi }{2009}\right)\sin\left(\frac{2020\pi }{2009}\right)\cdots\sin\left(\frac{4016\pi }{2009}\right)\right)$$

Note that

$$\sin\left(\frac{2012\pi }{2009}\right)=\sin\left(2\pi-\frac{2006\pi }{2009}\right)=-\sin\left(\frac{2006\pi }{2009}\right)$$

and

$$\sin\left(\frac{2016\pi }{2009}\right)=\sin\left(2\pi-\frac{2002\pi }{2009}\right)=-\sin\left(\frac{2002\pi }{2009}\right)$$ and so on and so forth,

it's obvious that

$$PQ=\frac{1}{2^{1004}}(-1)^{502}\sin\left(\frac{2\pi }{2009}\right)\sin\left(\frac{4\pi }{2009}\right)\sin\left(\frac{6\pi }{2009}\right)\cdots\sin\left(\frac{2008\pi }{2009}\right)$$

$$PQ=\frac{1}{2^{1004}}Q$$

$$\therefore P=\frac{1}{2^{1004}}$$

P.S. You might want to read this thread as well!http://mathhelpboards.com/trigonometry-12/simplify-cos-cos-2a-cos-3a-cos-999a-if-%3D-2pi-1999-a-253.html

I am unable to access the link above.

 

[MarkFL]

I am unable to access the link above.

Try this link:

http://mathhelpboards.com/trigonometry-12/simplify-cos-cos-2a-cos-3a-cos-999a-if-2pi-1999-a-253.html