$$\mathcal{F}(sin(t))=−i\pi(\delta(\omega−1)−\delta(\omega+1))$$ $$\mathcal{L}\{\ sin( t) \ \} \ \rightarrow \ doesn't\ exist\ ?$$

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$$\begin{equation*}\mathcal{F}\{\ sin( t) u( t) \ \} =−\frac{1}{2} \ i\pi \delta ( \omega −1) +\frac{1}{2} \ \ iπ\delta ( \omega +1) −\frac{1}{2} \ \ ( \omega −1) +\frac{1}{2} \ ( \omega +1)\end{equation*}$$ $$\begin{gather*}\mathcal{L}\{\ sin( t) u( t) \ \} \ =\frac{1}{s^{2} +1} \ ;\ Re\{s\} >0\\\\when\ s=jw\ doesn't\ resolve\ to\ it's\ fourier\ transform?\end{gather*}$$