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Wolfram Alpha gives following results for $\cos(x \cdot \imath)$ and $\sin(x \cdot \imath)$ ,where $x\in\mathbb{R}$: $\cos(x \cdot \imath)=\cosh(x)$ $\sin(x \cdot \imath)=\sinh(x)\cdot \imath$ What is a reason why the first number is real while the second is complex ? Definition of the $\cosh(x)$ and $\sinh(x)$ may be found here. |
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Note that, using Euler's formula, we can conclude that $$\cos(x)=\frac{e^{ix}+e^{-ix}}{2}$$ and $$\sin(x)=\frac{e^{ix}-e^{-ix}}{2i}.$$ Thus, $$\cos(ix)=\frac{e^{-x}+e^{x}}{2}=\cosh(x)$$ and $$\sin(ix)=\frac{e^{-x}-e^{x}}{2i}=i\left(\frac{e^{x}-e^{-x}}{2}\right)=i\sinh(x),$$ and it is clear from the definitions that $\sinh(x)$ and $\cosh(x)$ are both real for any $x\in\mathbb{R}$. Thus, $\cos(ix)$ is real for any $x\in\mathbb{R}$, and $\sin(ix)$ is imaginary for any $x\in\mathbb{R}$. |
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