Results 1 to 4 of 4

Math Help - cosx-sinx

  1. #1
    Newbie
    Joined
    Jul 2008
    Posts
    2

    cosx-sinx

    Hi Everbody lost my basic maths books was wondering how to

    convert cosx-sinx into a single sine function
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Dec 2007
    From
    Anchorage, AK
    Posts
    276

    Sum to product identity

    You can use one of the sum-to-product identities.
    For example,
    use \sin\theta=\cos\left(\frac{\pi}{2}-\theta\right)
    along with
    \cos{x}-\cos{y}=-2\sin\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right)
    To get
    \cos{x}-\sin{x}=\cos{x}-\cos\left(\frac{\pi}{2}-x\right)
    =-2\sin\left(\frac{x+\left(\frac{\pi}{2}-x\right)}{2}\right)\sin\left(\frac{x-\left(\frac{\pi}{2}-x\right)}{2}\right)
    =-2\sin\left(\frac{\pi}{4}\right)\sin\left(\frac{2x-\frac{\pi}{2}}{2}\right)
    =-\sqrt{2}\sin\left(x-\frac{\pi}{4}\right)
    Or, using \sin(\theta+\pi)=-\sin\theta,
    \cos{x}-\sin{x}=\sqrt{2}\sin\left(x-\frac{\pi}{4}+\pi\right)
    =\sqrt{2}\sin\left(x+\frac{3\pi}{4}\right).

    --Kevin C.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,686
    Thanks
    617

    Lightbulb

    Hello, hoops2008!

    Another approach . . .


    Convert \cos x -\sin x into a single sine function
    Multiply by \frac{\sqrt{2}}{\sqrt{2}}\!:\;\;\frac{\sqrt{2}}{\s  qrt{2}}\,(\cos x - \sin x) \;=\;\sqrt{2}\left(\frac{1}{\sqrt{2}}\cos x - \frac{1}{\sqrt{2}}\sin x\right)


    Note that: . \bigg\{\begin{array}{c}\sin\frac{\pi}{4} \:=\:\frac{1}{\sqrt{2}} \\ \cos\frac{\pi}{4} \:=\:\frac{1}{\sqrt{2}} \end{array}

    So we have: . \sqrt{2}\,\left(\sin\frac{\pi}{4}\cos x - \cos\frac{\pi}{4}\sin x\right) \;=\;\sqrt{2}\,\sin\left(\frac{\pi}{4} - x\right)<br /> <br />

    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Jul 2008
    Posts
    2
    Thanks for the replys fellas. I loved doing this kind of maths a long time ago and i am now very rusty.

    Could someone please verify that

    root2*415cos(100piet)-root2*415sin(100piet)=2*root830sin(wt- Pie/2)

    i couldnt get the maths symbols to copy over hence why i had the write "pie" and "root".

    this was the original question that my nephew asked me.

    Id be very greatful for some help here

    Thanks guys
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. sinx+cosx=0
    Posted in the Trigonometry Forum
    Replies: 5
    Last Post: July 24th 2011, 07:21 AM
  2. sin a sinx+ b cosx
    Posted in the Trigonometry Forum
    Replies: 1
    Last Post: November 23rd 2009, 07:48 AM
  3. sin a sinx+ b cosx
    Posted in the Trigonometry Forum
    Replies: 1
    Last Post: November 19th 2009, 07:35 AM
  4. Verify that √((1-cosx)/(1+cosx)) = (1-cosx)/|sinx|
    Posted in the Trigonometry Forum
    Replies: 3
    Last Post: December 10th 2008, 08:39 PM
  5. cosx and sinx when x= 0
    Posted in the Trigonometry Forum
    Replies: 1
    Last Post: October 1st 2008, 05:26 PM

Search Tags


/mathhelpforum @mathhelpforum