Question

Good Question

Answer 1

\[\huge y = \cos(i \theta) + i \sin(i \theta)\]
prove that y ϵ R

Answer 2

Use the definitions \[\sin(z)=\frac{e ^{iz}-e ^{-iz}}{2i}~~~\cos(z)=\frac{e ^{iz}+e ^{-iz}}{2}\]Substitute i theta for z, and it will be evident that both terms are real numbers.

Answer 3

since : \[\huge e^{i \theta } = \cos(\theta) + isin(\theta)\]
therefore: \[\huge \cos(i \theta) + isin(i \theta) = e^{i i \theta} = e^{- \theta} = ({1 \over e})^{\theta} \]
and \[\huge e \ and \ \theta \in \mathbb{R} \]
so \[\huge y \in \mathbb{R}\]