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Simplify an equation

  1. Mar 10, 2017 #1
    1. The problem statement, all variables and given/known data
    upload_2017-3-10_23-8-47.png



    3. The attempt at a solution
    The way i have simplified it:
    upload_2017-3-10_23-9-58.png

    I do not know it is the way to do it.

    But if so, now I need to give the answer in cartesian form, and I think I need to use the formula:

    upload_2017-3-10_23-12-0.png
    meaning it will look like:

    upload_2017-3-10_23-12-35.png

    Anyone who can tell me if I am on the right track?
    Thank you
     
  2. jcsd
  3. Mar 10, 2017 #2

    Mark44

    Staff: Mentor

    What you have is not an equation... An equation always has an = sign between two expressions.
    You're on the right track, based on a quick scan, but you're not done. Cartesian form means a + bi (or a + jb, as you are writing things.)

    One thing, though -- the first image (which by the way we discourage) has what appears to be the magnitude of the numerator. Because you posted an image, I can't copy just the numerator to show you what I mean.

    One other thing -- the work in the inages is very neat, but not well organized. You start off with an expression that you're supposed to simplify, but you have other things in there that are elaborations of parts of what you're working on, and there are indications that what you're starting with and what you're ending with are actually equal. The = symbol is used for that purpose.
     
  4. Mar 10, 2017 #3
    What is j ? ##j = \sqrt{-1}## ?

    Second step is wrong.

    ##|(1- 2j)(1/\sqrt{2}) + (1- 2j)j(1/\sqrt{2})|/...##.
     
  5. Mar 10, 2017 #4
    First of all, thank you for your feedback.
    I know that I am not done, as you say I need to find the answer in Cartesian form in Denmark (a+jb), that is why I am asking if i have to use the formula:
    upload_2017-3-10_23-12-0-png.114359.png
    which I am confirmed that I have to.

    And I can see, what you mean with the equal sign. Thank you.
     
  6. Mar 10, 2017 #5

    Ray Vickson

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    Do you know the result ##|z_1 z_2| =|z_1| \, |z_2| ?## Just apply it in the numerator, with ##z_1 = 1 - 2 j## and ##z_2 = e^{j \pi/4}##.
     
  7. Mar 10, 2017 #6

    Mark44

    Staff: Mentor

    Rather than rely on a formula (that by the way is incorrect), the way that division of complex numbers is to multiply by 1 in a suitable form.
    If ##z_1 = x_1 + jy_1## and ##z_2 = x_2 + jy_2##, then
    ##\frac{z_1}{z_2} = \frac{x_1 + jy_1}{x_2 + jy_2} = \frac{x_1 + jy_1}{x_2 + jy_2} \cdot \frac{x_2 - jy_2}{x_2 - jy_2}##
    ##= \frac{x_1x_2 + y_1y_2}{x_2^2 + y_2^2 } + j\frac{x_2y_1 - x_1y_2}{x_2^2 + y_2^2}##
    In the next to last step, I am multiplying by 1, in the form of ##x_2 - jy_2## over itself. The reason for multiplying by ##x_2 - jy_2## over itself is that this is the complex conjugate of what was originally the denominator, ##x_2 + jy_2##.

    The result is now in Cartesian form, with the first fraction being the real part, and the stuff after j being the imaginary part. In your formula, you omitted j.
     
  8. Mar 11, 2017 #7
    Will it then be:

    x1= upload_2017-3-11_16-48-32.png
    y_1 = upload_2017-3-11_16-49-30.png
    x_2 = upload_2017-3-11_16-50-1.png
    y_2 = upload_2017-3-11_16-50-29.png

    so:

    = upload_2017-3-11_16-53-26.png

    in cartesian I get

    = 1.7134 +j1.7134

    I really hope this is correct.
     

    Attached Files:

    Last edited by a moderator: Mar 11, 2017
  9. Mar 12, 2017 #8

    SammyS

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    I get something much different.
     
  10. Mar 12, 2017 #9

    haruspex

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    2016 Award

    Those are both wrong. Please post the steps by which you arrived at them.
     
  11. Mar 13, 2017 #10

    Ray Vickson

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    Science Advisor
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    As far as I can see you are on the wrong track. Your expression is of the form ##N/D##, where the numerator is ##N = |(1-j2) e^{j \pi/4}| ## and the denominator is
    $$D = \sinh (j \pi/6) + 3 (\cos(\pi/6) + j \sin(\pi/6)).$$
    I assume that ##j2## in the numerator means ##2j##, not ##j^2##.

    When evaluating ##N## you first expanded ##e^{i \pi/4}##, then multiplied by ##(1-2j)##, but you did it all wrong (at least as written and read using standard rules). You should have
    $$(1-2j) ((\sqrt{2}/2 + j \sqrt{2}/2) = (1-2j) \sqrt{2}/2 + (1-2j) j \sqrt{2}/2\\ = 3 \sqrt{2}/2- j \sqrt{2}/2 \doteq 2.121 - 0.707 j ,$$
    but you have written
    $$(1-2j) \sqrt{2}/2 + j \sqrt{2}/2 = \sqrt{2}/2 - j \sqrt{2}/2 \doteq 0.707 - 0.707 j.$$
    You are bound to get the wrong answer!

    However, all that is a real waste of time; just pay attention to post #5, and use the much simpler result ##N = |1-2j| |e^{j \pi/4}|.## Can you take it from there?
     
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