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Not sure if I have done this correctly, seems too straight forward, any help is very appreciated.

QUESTION:
Find the real and imaginary parts of $f(z) = \cos(z)$.

ATTEMPT:
$\cos(z) = \cos(x+iy) = \cos x\cos(iy) − \sin x\sin(iy) = \cos x\cosh y − i\sin x\sinh y$

Is that correct?

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3 Answers

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By definition, $$ \cos z=\frac{e^{iz}+e^{-iz}}{2},\qquad \sin z=\frac{e^{iz}-e^{-iz}}{2i} $$ In particular, for real $y$, $$ \cos(iy)=\frac{e^{-y}+e^{y}}{2}=\cosh y $$ and $$ \sin(iy)=\frac{e^{-y}-e^{y}}{2i}=i\frac{e^{y}-e^{-y}}{2}=i\sinh y $$

So, yes, you're correct.

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It is simple, but tis the beauty of the trig/exponential functions! You're $correct$!

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Using the exponential definition of the cosine,

$$2\cos(z)=e^{iz}+e^{-iz}=e^{-y+ix}+e^{y-ix}\\ =e^{-y}(\cos(x)+i\sin(x))+e^{y}(\cos(x)-i\sin(x))\\ =(e^y+e^{-y})\cos(x)-i(e^y-e^{-y})\sin(x)).$$

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