Results 1 to 6 of 6

Thread: Use cos(Z) = (e^[iz] + e^[-iz]) / 2 to solve cos(Z) = 4

  1. #1

    Use cos(Z) = (e^[iz] + e^[-iz]) / 2 to solve cos(Z) = 4

    Problem needing assistance: complex numbers

    Use definition cos Z =(e[iz] + e[-iz])/2 to find 2 imaginary numbers having a cosine of 4.


    Please note that the iz and the -iz are exponents.

    This is for the class of IB Higher Level Math 3.

    Thanks!

  2. #2
    Elite Member
    Join Date
    Jun 2007
    Posts
    12,871

    Re: Complex Numbers - Help

    Quote Originally Posted by adele.fielding
    Problem needing assistance: complex numbers

    Use definition cos Z =(e[iz] + e[-iz])/2 to find 2 imaginary numbers having a cosine of 4.


    This is for the class of IB Higher Level Math 3.

    Thanks!
    Please share with us your work - indicating exactly where you are stuck - so that we know where to begin to help you.
    “... mathematics is only the art of saying the same thing in different words” - B. Russell

  3. #3

    Re: Complex Numbers - Help

    replaced cos z by 4, expanded the exponents for eto iz = cos z + i sin z took us in circles back to cos z = 4.

    Not getting anywhere.

  4. #4
    Elite Member
    Join Date
    Jun 2007
    Posts
    12,871

    Re: Complex Numbers - Help

    Quote Originally Posted by adele.fielding
    replaced cos z by 4, expanded the exponents for eto iz = cos z + i sin z took us in circles back to cos z = 4.

    Not getting anywhere.
    [tex]\frac{e^{iz} - e^{-iz}}{2} \, = \, 4[/tex]

    [tex]e^{iz} \, - \, e^{-iz} \, = \, 8[/tex]

    [tex]e^{iz} \, - \, \frac{1}{e^{iz}} \, = \, 8[/tex]

    This reduces to quadratic equation - solve....
    “... mathematics is only the art of saying the same thing in different words” - B. Russell

  5. #5

    Re: Complex Numbers - Help

    Thankyou so much, that was all we needed to complete the problem.

    Can you also explain (or provide online reference) how it is that the cos z can have a value of 4? In real numbers, cos x is between plus and minus 1. How should we be thinking about cos z in complex plane?

    Thanks,
    aF

  6. #6
    Elite Member
    Join Date
    Jan 2005
    Posts
    6,327

    Re: Complex Numbers - Help

    I don’t know what your course level. But I will answer your question in a basic way.
    The complex function [tex]\cos (z) = \cos (x)\cosh (y) - i\sin (x)\sinh (y)[/tex], where [tex]z=x+yi[/tex].
    So if [tex]\cos(z)=4[/tex] we must have [tex]\cos (x)\cosh (y)=4[/tex] and [tex]\sin (x)\sinh (y)=0[/tex].
    That happens if [tex]y=0[/tex] but that means [tex]\cos (x)=4[/tex] which is impossible.
    Thus we must have [tex]x=0[/tex] or a even multiple of [tex]\pi[/tex] which gives [tex]\cosh(y) = 4\,\& \,y = \frac{{8 \pm \sqrt {68} }}{2}[/tex].
    [tex]z= i \frac{{8 \pm \sqrt {68} }}{2}[/tex].
    “A professor is someone who talks in someone else’s sleep”
    W.H. Auden

Bookmarks

Posting Permissions

  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •