Question 24.13: Find F{sin at}....

Subtracting the previous expressions for  {\mathcal{F}}\{\mathrm{e}^{j t a}\}  and   {\mathcal{F}}\{\mathrm{e}^{-j t a}\}  and using Euler’s relations we find

{\mathcal{F}}\{\mathrm{e}^{jt a}\}-{\mathcal{F}}\{\mathrm{e}^{-j t a}\}=2\pi(\delta(\omega-a)-\delta(\omega+a))

that is,

{\mathcal{F}}{\biggl\{}{\frac{\mathrm{e}^{\mathbf{j}ta}-\mathbf{e}^{-\mathbf{j}t a}}{2\mathbf{j}}}{\biggr\}}={\frac{\pi}{\mathbf{j}}}(\delta(\omega-a)-\delta(\omega+a))

so that

{\mathcal{F}}\{\sin a t\}={\frac{\pi}{{\mathfrak{j}}}}(\delta(\omega-a)-\delta(\omega+a))