1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Use exponential notation to form a+ib

  1. Mar 5, 2013 #1
    1. The problem statement, all variables and given/known data
    Use exponential notation to write (√3-i)(1+i√3) in the form a+ib.


    2. Relevant equations



    3. The attempt at a solution
    Let (√3-i) = z1
    r1=|z1|=√((√3)2-i2)
    =√(3-1)
    =√2

    Therefore,
    √3=√2cosθ and -1=√2sinθ
    cosθ = √3/√2 and sinθ = -1/√2

    θ= arcsin(-1/√2)
    θ= -∏/4

    z1 = √2(cos(-∏/4)+isin(-∏/4))

    Let (1+i√3) = z2
    r2 = |z2| = √(12+(i√3)2)
    = √(1+(1(3)) <-- I am very uncertain about this step
    = √4
    = 2

    Therefore
    1 = 2cosθ and 3 = 2sinθ
    cosθ = 1/2 and sinθ = 3/2

    However,
    θ=arcsin(3/2) does not compute which leads me to believe that (i√3)2 = 3 is incorrect. Any help is appreciated. Thank you.
     
  2. jcsd
  3. Mar 5, 2013 #2

    SammyS

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    What is i2 ?
     
  4. Mar 5, 2013 #3
    i2 = -1
    However if I plug that in, I get this.
    r2 = |z2| = √(12+(i√3)2}
    = √(1+(-1(3))
    = √-2
    Which still does not compute.
     
  5. Mar 5, 2013 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    |a+bi|=sqrt(a^2+b^2).
     
  6. Mar 5, 2013 #5
    Ah, right. This leads me to another question. Do I use the "1" from i and eliminate the sqrt3, or treat it as (sqrt3)i and use the resulting 3? I would guess the latter.
     
  7. Mar 5, 2013 #6

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    I'm not sure what you are asking. If z2=1+i*sqrt(3) then a=1 and b=sqrt(3). sqrt(a^2+b^2)=sqrt(1+3)=2.
     
  8. Mar 5, 2013 #7
    That is what I was asking and what I thought. Thank you. Now let's see if I can wrestle this into exponential form.
     
  9. Mar 5, 2013 #8

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Good. Your probably know using exponential form is a complete waste of time, right? You could just multiply (√3-i)(1+i√3) out and be done with it. But if that's what the ask you to do then that's what you should do. It'll be easy to check your answer.
     
  10. Mar 5, 2013 #9
    Well, the problem states "Use exponential notation to write (√3-i)(1+i√3) in the form a+ib." Have I been wasting my time?
     
  11. Mar 5, 2013 #10

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Not if the problem forces you to do it that way. I'm just saying you can easily check your answer by doing it the simple way.
     
  12. Mar 5, 2013 #11
    z1 = √2{cos(-∏/4)+isin(-∏/4)} = √2ei(-∏/4)

    z2 = 2{cos∏/3+isin∏/3) = 2ei∏/3

    z1z2 = 2√2ei(-∏/4)+i∏/3
    =2√2ei∏/12

    So far so good?
     
  13. Mar 5, 2013 #12

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    z2=2*exp(i*pi/3), that's good. z1 isn't. You might be rehashing some of your old bad stuff into that one.
     
  14. Mar 5, 2013 #13
    z1 = 2{cos11∏/6+isin11∏/6}
    = 2ei11∏/6
     
  15. Mar 5, 2013 #14

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    I would have said 2exp(-i*pi/6), but sure, that's the same as 2exp(i*11pi/6). Think you are almost there.
     
  16. Mar 5, 2013 #15
    z1z2 = 2*2ei11∏/6+i∏/3
    = 4ei13∏/6

    4{cos13∏/6+isin13∏/6} Trig confuses me, so let's see.
    4{-cos∏/6-isin∏/6}
    4(-√3/2)-(i/2)
    -2√3-2i
     
  17. Mar 5, 2013 #16

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    4*exp(13*pi/6) is correct. I guess trig does confuse you. cos(13*pi/6) isn't equal to -cos(pi/6). Why would you think that?
     
  18. Mar 5, 2013 #17
    I don't know what I did there. Does it instead behave the same as ∏/6? Making my answer
    2√3+2i?
     
  19. Mar 5, 2013 #18

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Sure, multiply out (√3-i)(1+i√3) to confirm that.
     
  20. Mar 5, 2013 #19
    Thank you.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Use exponential notation to form a+ib
  1. Exponential form (Replies: 6)

Loading...