The function $(-1)^{-x}$

First you need to assume that you're working within the complex numbers, meaning that we assume $-1\in \mathbb C$ - otherwise your expression is simply not defined - and further we assume that $x\in \mathbb{R}$.

Now we perform some basic manipulations on your expression and get $$ (-1)^{-x}=\exp\{\log((-1)^{-x})\}=\exp\{-x\log(-1)\} $$ The expression $\log(-1)$ is not defined in $\mathbb R$ but is very much in $\mathbb C$, we are working with the complex logarithm and are using the principal value, we get $$ \log(-1)=\log(1)+i\mathrm{arg}(1)=i\pi $$ Now we put everything back together and we get using the Euler formula for $x\in \mathbb R$ \begin{align} (-1)^{-x}=\exp\{\log((-1)^{-x})\}=&\exp\{-x\log(-1)\}\\ =&\exp\{-xi\pi\}\\ =&\cos(x\pi)-i\sin(\pi x) \end{align} Now you can plot your function, decomposed into an imaginary and real part, and you'll get the desired $\sin(\cdot),\cos(\cdot)$ functions, which are scaled accordingly. That means $$ \Re((-1)^{-x})=\cos(x\pi) \text{ and } \Im((-1)^{-x})=-\sin(\pi x) $$

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