Can we apply Euler's formula on plain waves?

Question

I don't speak English so edit my question if it is not accurate.

Euler's formula for a complex number is: $$e^{i\theta}=\cos\theta + i \sin\theta$$ But when I write a plain wave as $$e^{i\vec{k}\cdot\vec{x}}=\cos(\vec{k}\cdot\vec{x}) + i \sin(\vec{k}\cdot\vec{x})$$ I think something is not right because $\vec{k}\cdot\vec{x}=kx\cos(\beta)$ that means we are taking the cosine of another cosine: Setting $kx=1$ we have $$e^{i\cos(\beta)}=\cos(\cos(\beta)) + i \sin(\cos(\beta))$$ which is a bit non-intuitive. So, I thought you may help me understanding $e^{i\vec{k}\cdot\vec{x}}$. Can we apply Euler's formula on plain waves? How did we extend an angle to a dot product?

Answer 1

Setting $kx=1$ is probably what confuses you. If you go to a Cartesian frame you have $\vec{k}\cdot \vec{x} = k_x x + k_y y + k_z z$ which is a number. Let us call this number $\ell$. We then have $$ e^{i\vec{k}\cdot \vec{x}}= e^{i\ell} = \cos \ell + i \sin \ell $$ So your formula is perfectly legitimate.