Determine x coordinate on a sine wave

Question

I have this question which is puzzling me:

I understand that to determine the x value from the equation y = sin(x) you would do x = arcsin(y) but how would i use this to determine the x value of b?

Thanks

Answer 1

A good way to understand the "multivalued" aspect of $\sin^{-1}$ is to reverse the graphical representation of $y=\sin(x)$ and consider "layers" of two types (see figure below with a color code green/red):

$$\begin{cases} y&=&\ \ \ \color{green}{\sin(x)}+2k \pi& \text{or}\\ y&=&\color{red}{-\sin(x)}+(2k+1) \pi \end{cases} \ \ \ (\text{for integer values of } \ k )$$

with two "mother curves": one for $y=\sin^{-1}(x)$ (well recognizable) and the other one $y=-\sin(x)$ (dotted curve)