Results 1 to 4 of 4

Math Help - sum of cosines

  1. #1
    Member
    Joined
    Dec 2008
    Posts
    86

    sum of cosines


    I have the function: y(t) = (cos(300πt) + sin(500πt))^3

    I need too expand the expression of y to get a sum of cosines with positive frequencies. Using
    trigonometric identities find the frequencies of resulting sine waves.Can someone help me with that?
    Thank you

    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,392
    Thanks
    56
    Quote Originally Posted by qwerty321 View Post
    I have the function: y(t) = (cos(300πt) + sin(500πt))^3

    I need too expand the expression of y to get a sum of cosines with positive frequencies. Using
    trigonometric identities find the frequencies of resulting sine waves.Can someone help me with that?
    Thank you
    Why only cosine terms? Why not sine and cosine term?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Dec 2008
    Posts
    86
    because what I have to do is:
    Expand the expression of
    y(t) to get a sum of cosines with positive frequencies. Using trigonometric identities find the frequencies of resulting sine waves and compare with the frequency components obtained using MATLAB

    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Grandad's Avatar
    Joined
    Dec 2008
    From
    South Coast of England
    Posts
    2,570
    Thanks
    1

    Sum of cosines

    Hello qwerty321
    Quote Originally Posted by qwerty321 View Post
    Quote Originally Posted by qwerty321 View Post
    I have the function: y(t) = (cos(300πt) + sin(500πt))^3

    I need too expand the expression of y to get a sum of cosines with positive frequencies. Using
    trigonometric identities find the frequencies of resulting sine waves.Can someone help me with that?
    Thank you

    Using A to stand for 300\pi t and B for 500 \pi t:

    (\cos A + \sin B)^3 = \cos^3A+3\cos^2A\sin B + 3\cos A \sin^2B+\sin^3B

    Now make repeated use of the following identities:

    \cos^2x = \frac{1}{2}(\cos 2x -1)

    \sin^2x = \frac{1}{2}(1-\cos 2x)

    \cos x \cos y = \frac{1}{2}(\cos(x+y) + \cos(x-y))

    \sin x \cos y = \frac{1}{2}(\sin(x+y) + sin(x-y))

    So, for example: \cos^3A = \frac{1}{2}(\cos 2A -1)\cos A

    = \frac{1}{2}\left(\frac{1}{2}(\cos3A + \cos A) -\cos A\right)

    = \frac{1}{4}(\cos 3A - \cos A)

    And the second term is: 3 \cos^2 A \sin B = \frac{3}{2}(\cos 2A -1) \sin B

    = \frac{3}{2}\left(\frac{1}{2}(\sin(B+2A)+\sin(B-2A)) -\sin B \right)

    Similarly with the last two terms. Finally, if you need to get an expression in terms of cosine only, you could use \sin x = \cos(\pi /2 - x).

    Grandad
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Law of Cosines
    Posted in the Trigonometry Forum
    Replies: 1
    Last Post: November 19th 2009, 12:52 PM
  2. sum of cosines
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 31st 2009, 05:40 PM
  3. Law of Cosines
    Posted in the Trigonometry Forum
    Replies: 1
    Last Post: August 13th 2009, 07:49 AM
  4. Help - Sum of Cosines
    Posted in the Calculus Forum
    Replies: 3
    Last Post: May 25th 2009, 05:50 PM
  5. The Law of Cosines
    Posted in the Trigonometry Forum
    Replies: 1
    Last Post: April 28th 2005, 12:29 PM

Search Tags


/mathhelpforum @mathhelpforum