1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Complex tangent formula proof

  1. Jul 6, 2015 #1
    1. The problem statement, all variables and given/known data

    This is an easy one, but keep in mind I'm kind of a newbie, anyway I can't figure out how to get the next formula...
    tan(z) = (tan(a)+i tanh(b))/(1 - i tan(a)tan(b))

    2. Relevant equations

    This is the third part of an excercise, previous I proof the follow, -all using the definitions, of complex sin,cos, and tan, and definitions of real sinh,cosh, and tanh-...

    cos(z)=cos(a)cosh(b)-i sin(a)sinh(b);

    3. The attempt at a solution

    I tried a lot of things, but couldn't get any way, maybe I'm missing something important, and that's what I fear.
    I tried to replace tan(z) = sin(z)/cos(z) with the other equations but, I'm getting nothing.

    Sorry for my english, lot of thanks for the help!
  2. jcsd
  3. Jul 6, 2015 #2


    User Avatar
    Science Advisor

    [itex] \tan(z)=\frac{\sin(z)}{\cos(z)}=\frac{\frac{e^{z}-e^{-z}}{2i}}{\frac{e^{z}+e^{-z}}{2}}=\frac{1}{i}\frac{e^{z}-e^{-z}}{e^{z}+e^{-z}}=\frac{1}{i}\frac{e^{a+ib}-e^{-(a+ib)}}{e^{a+ib}+e^{-(a+ib)}}[/itex]
    Now write out the expression for [itex]e^{a+ib} [/itex]...
  4. Jul 6, 2015 #3
    Thanks for that!, but, I will have to ask for even more help. Because, I already knew that formula, and I cant find how to pass to the one I'm asking for...

    [itex]tan(z) = (tan(a)+i tanh(b))/(1 - i tan(a)tan(b))[/itex]
  5. Jul 6, 2015 #4
    I've solved it! and I wanted to tell you. The biggest problem was that the damn book have a cute problem tan(z)=(tan(a)+itanh(b))/(1−itan(a)tan(b)) is really tan(z)=(tan(a)+itanh(b))/(1−itan(a)tanh(b)) having that in mind and with some equations it's really simple to get. I mean, using cosh2(x)- sinh2(x)=1,cos2(x)+sin2(x)=1, and tan(x)=sin(x)/cos(x), ....and the most important, after the two other equations I posted at first... (and the simplest one) sin(x)=cos(x)tan(x)
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted