Solve y=-1/2sin(2pix-pi/2)+3

$$y = - \frac { 1 } { 2 } \sin ( 2 \pi x - \frac { \pi } { 2 } ) + 3$$

Solve for x (complex solution)

$$x=-\frac{i\ln(-i\sqrt{4y^{2}-24y+35}-2iy+6i)}{2\pi }+n_{1}+\frac{1}{4}\text{, }n_{1}\in \mathrm{Z}$$

$$x=-\frac{i\ln(i\sqrt{4y^{2}-24y+35}-2iy+6i)}{2\pi }+n_{2}+\frac{1}{4}\text{, }n_{2}\in \mathrm{Z}$$

Solve for y (complex solution)

$$y=\frac{ie^{2\pi ix-\frac{\pi i}{2}}}{4}-\frac{ie^{-2\pi ix+\frac{\pi i}{2}}}{4}+3$$

Solve for x

$$x=\frac{\frac{2\arcsin(2\left(3-y\right))}{\pi }+4n_{1}+1}{4}\text{, }n_{1}\in \mathrm{Z}$$

$$x=\frac{-\frac{2\arcsin(2\left(3-y\right))}{\pi }+4n_{2}+3}{4}\text{, }n_{2}\in \mathrm{Z}\text{, }y\geq \frac{5}{2}\text{ and }y\leq \frac{7}{2}$$

Solve for y

$$y=\frac{-\sin(\frac{\pi \left(4x-1\right)}{2})+6}{2}$$