Results 1 to 2 of 2

Math Help - Sine and Cosine of i (imaginary unit)

  1. #1
    Newbie
    Joined
    Mar 2008
    Posts
    2

    Sine and Cosine of i (imaginary unit)

    The following is my working for the Sine and Cosine of the imaginary unit, i.

    e^{ix}=cos(x)+i(sin(x))
    Substituting x=i gives
    e^{-1}=cos(i)+i(sin(i))
    Squaring both sides
    e^{-2}=(cos(i))^2-(sin(i))^2+2i(sin(i)cos(i))
    e^{-2}=(1-2((sin(i))^2)+2i(sin(i)cos(i))
    Rearranging the original equation in terms of cos(i) gives
    cos(i)=e^{-1}-i(sin(i))
    Substituing this back into e^{-2}=(1-2((sin(i))^2)+2i(sin(i)cos(i)) gives
    e^{-2}=(1-2((sin(i))^2)+2i(sin(i))(e^{-1}-i(sin(i)))
    e^{-2}=(1-2((sin(i))^2)+2i(sin(i))(e^{-1}) -2(i^2)((sin(i))^2)
    e^{-2}=1+2i(sin(i))(e^{-1})
    Rearranging for sin(i) gives
    (e^{-2}-1)/2i(e^{-1})=sin(i)
    sin(i)=(1-e^2)/2ei
    Then to find Cos(i): Substituting back into the original equation gives
    cos(i)=e^{-1}-(i(1-e^2))/(2ei)<br />
cos(i)=(1-e^2)/2e

    The problem is that all the sources I can find for the sine and cosine of i on the internet say
    sin(i) = ((e-e^{-1})/2)i or ((e^2-1)/2e)i and
    cos(i)=(e+e^{-1})/2 or ( e^2+1)/2e .

    As you can see my answers are very close to the others, I just can't figure out where the difference in signs comes from... also, is there any difference if the i is put outside the fraction like they have, or on the bottom like I work it to be?

    If I've made any typos or not made anything clear enough, just point it out and I'll change the original post.

    Thanks, Spud.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Oct 2007
    From
    London / Cambridge
    Posts
    591
    e^{ix} = \cos x + i \sin x   \ \ \ \ (1)
    e^{-ix} = \cos x - i \sin x   \ \ \ (2)

    (1) + (2) gives

    e^{ix} + e^{-ix} = 2 \cos x

    \cos x = \frac{1}{2} \left ( e^{ix} + e^{-ix} \right )

    \cos i = \frac{1}{2} \left ( e^{-1} + e^{1} \right )

    I will leave you to do \sin i .

    - Bobak
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. natural log and imaginary unit question
    Posted in the Pre-Calculus Forum
    Replies: 5
    Last Post: November 27th 2009, 01:26 PM
  2. real number value for imaginary unit
    Posted in the Algebra Forum
    Replies: 2
    Last Post: September 28th 2009, 08:36 AM
  3. Imaginary unit
    Posted in the Math Topics Forum
    Replies: 4
    Last Post: March 26th 2009, 03:44 PM
  4. cosine and sine..
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 4th 2008, 01:49 AM
  5. Sine and Cosine of i (imaginary unit)
    Posted in the Advanced Math Topics Forum
    Replies: 1
    Last Post: March 2nd 2008, 08:08 AM

Search Tags


/mathhelpforum @mathhelpforum