Results 1 to 5 of 5

Math Help - limiting value of Cosine

  1. #1
    Member
    Joined
    Feb 2009
    From
    Chennai
    Posts
    148

    limiting value of Cosine

    Hi--

    Please help me as to what value does \cos(\theta) \times \cos(\theta/2) \times \cdots \times \cos(\theta/2^{n}) approach as n \to \infty.
    Last edited by mr fantastic; June 28th 2010 at 02:22 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
    Joined
    Dec 2009
    From
    Russia
    Posts
    1,506
    Thanks
    1
    Hint:

    Multiply (and divide) by 2sin\frac{\theta}{2} and use:

    sin2x=2sinxcosx
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Feb 2009
    From
    Chennai
    Posts
    148
    I was solving an analysis problem and i need this quantity to have 1/2^{n} multiplied by some value. If this doesn't work out then my claim would be incorrect.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by Chandru1 View Post
    Hi--

    Please help me as to what value does \cos(\theta) \times \cos(\theta/2) \times \cdots \times \cos(\theta/2^{n}) approach as n \to \infty.
    This infinite product is based on a famous infinite product first discovered by Euler:

    \sin (x) = 2 \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right)

     = 2^2 \sin \left(\frac{x}{4}\right) \cos  \left(\frac{x}{4}\right) \cos \left(\frac{x}{2}\right)

     = 2^3 \sin \left(\frac{x}{8}\right) \cos  \left(\frac{x}{8}\right) \cos  \left(\frac{x}{4}\right) \cos \left(\frac{x}{2}\right)

    = ....

    = 2^n \sin \left(\frac{x}{2^n}\right) \cos  \left(\frac{x}{2^n}\right) .... \cos \left(\frac{x}{2}\right)

    = x \cdot \left[ \frac{ \sin \left(\frac{x}{2^n}\right) }{\frac{x}{2^n}}  \right] \cos  \left(\frac{x}{2}\right)  \cos  \left(\frac{x}{4}\right).... \cos \left(\frac{x}{2^n}\right).


    Now take the limit n \to \infty:

    \sin (x) = x \cos  \left(\frac{x}{2}\right)  \cos  \left(\frac{x}{4}\right).... \cos \left(\frac{x}{2^n}\right) ....

    I hope you can see what to do with this famous result.
    Last edited by mr fantastic; June 28th 2010 at 04:53 PM. Reason: Added ....
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Feb 2009
    From
    Chennai
    Posts
    148

    thanks

    Mr. Fantastic thanks for the fantastic answer.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: June 22nd 2011, 01:43 PM
  2. Why is cosine(-x) equal to cosine(x)?
    Posted in the Trigonometry Forum
    Replies: 4
    Last Post: August 6th 2010, 06:56 PM
  3. Limiting
    Posted in the Advanced Statistics Forum
    Replies: 4
    Last Post: July 9th 2010, 11:15 AM
  4. Limiting parallels
    Posted in the Geometry Forum
    Replies: 1
    Last Post: April 4th 2010, 11:17 PM
  5. What are the Limiting Distribution and Limiting Probabilities
    Posted in the Advanced Statistics Forum
    Replies: 4
    Last Post: April 3rd 2009, 01:49 PM

Search Tags


/mathhelpforum @mathhelpforum