Solve y=0.5sin(1.5x+1)

$$y = 0.5 \sin ( 1.5 x + 1 )$$

Solve for x (complex solution)

$$x=-\frac{2i\ln(i\sqrt{4y^{2}-1}+2iy)}{3}+\frac{4\pi n_{1}}{3}-\frac{2}{3}\text{, }n_{1}\in \mathrm{Z}$$

$$x=-\frac{2i\ln(-i\sqrt{4y^{2}-1}+2iy)}{3}+\frac{4\pi n_{2}}{3}-\frac{2}{3}\text{, }n_{2}\in \mathrm{Z}$$

Solve for y (complex solution)

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

Solve for x

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

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

Solve for y

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