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

[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.

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

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

Thanks!

 

[Deleted member 4993]

Re: Complex Numbers - Help

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.

 

[adele.fielding]

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.

 

[Deleted member 4993]

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.

\(\displaystyle \frac{e^{iz} - e^{-iz}}{2} \, = \, 4\)

\(\displaystyle e^{iz} \, - \, e^{-iz} \, = \, 8\)

\(\displaystyle e^{iz} \, - \, \frac{1}{e^{iz}} \, = \, 8\)

This reduces to quadratic equation - solve....

 

[adele.fielding]

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

 

[pka]

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 \(\displaystyle \cos (z) = \cos (x)\cosh (y) - i\sin (x)\sinh (y)\), where \(\displaystyle z=x+yi\).
So if \(\displaystyle \cos(z)=4\) we must have \(\displaystyle \cos (x)\cosh (y)=4\) and \(\displaystyle \sin (x)\sinh (y)=0\).
That happens if \(\displaystyle y=0\) but that means \(\displaystyle \cos (x)=4\) which is impossible.
Thus we must have \(\displaystyle x=0\) or a even multiple of \(\displaystyle \pi\) which gives \(\displaystyle \cosh(y) = 4\,\& \,y = \frac{{8 \pm \sqrt {68} }}{2}\).
\(\displaystyle z= i \frac{{8 \pm \sqrt {68} }}{2}\).