How can I find all w in C such that cos(z + w) = cos z for all z in C?

[Mark C]

Hi,

How would I do the following:

Find all w in C, such that cos(z + w) = cos z, for all z in C.

Of course I would need to use the identity cos z = (e^iz +e^-iz)/2, but I have trouble finding w for all z.

Thank you

 

[dextercioby]

Just to get to "warmed"...Can u solve the equation into reals??

Daniel.

 

[Zurtex]

Err I mess about with the expression a bit using trig laws and din't get very far.

But erm wouldn't the obvious be w = 2*pi and by extention all the others you should know but I've edited it out just to give you something to do.

 

[Mark C]

Yes, w=2(pi) is certainly one solution, and showing that w = 2(pi)n (n an integer), works for all z is easy. However, how could I know that there are no other w that do this? Meaning, I could also show w = 4(pi)n works for all z, but those are not all the w's possible.

Thanks

 

[dextercioby]

What's the range of the second "n",the one included in "4\pi n"??

If it's Z (like the one with "2 \pi n"),then you have to agree that in this case there's nor NEW solution...

Daniel.

 

[Mark C]

If I show, that w = 4(pi)n works for all z (n an integer), then I do not find all such w that do this, for all z in C, because 2(pi), works also, but is not in the set 4(pi)n , n an integer. So, if I would claim that all w=4(pi)n works for all z, this is true, but, I do not find all such w.